This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
O,n
= 0,1,2, ... ) ,
that lim M(r, f) r--+oo
= 00
rP
for each positive integer p. This implies lim log M(r, f) = log r
00 .
r--+oo
On the other hand, taking p = 2r in (1.33), we have log+ M(r, f) ~ 3T(2r, f) . Hence (1.38) holds.
(1.39)
15
Nevanlinna's Theory of Meromorphic Functions
20 f(z) has poles. First assume that f(z) has an infinite number of poles. Then from
N(r2, J)
~
N(r2, f) - N(r, f)
~
n(r, J) log r
(r > 1) ,
we have lim N(r, J) = log r
00
r-+oo
and, a fortiori, (1.38). Next assume that f(z) has only a finite number of poles bj(j = 1,2, ... ,k) whose orders are respectively mj(j = 1,2, ... ,k). Set k
P(z) =
II (z -
bj)mj,
g(z) = P(z)f(z) .
j=1 Then, remembering that f(z) is not a rational function, g(z) is a transcendental entire function, and hence (1.38) holds for g(z). On the other hand, by (1.30),
T(r, g)
~
T(r, P) + T(r, J)
~ mlogr
(r ~ 1) ,
+ K + T(r, J)
where m = E~=1 mj and K > 0 is a constant. Hence (1.38) also holds. Concerning the growth of a meromorphic function, an important notion is that of its order. The order p of a non-constant meromorphic function f(z) is defined by p = lim log T(r, J) . (1.40) r-+oo log r We have 0 ~ p ~ 00, and we shall denote it by p(J). When p is finite, then, for each positive number e, on one hand there is a value ro such that
T(r,J)
2(p-r),
1 1 log+ - - ~ log+ - - + log 2 . P - PI P- r
(1.55)
Finally from (1.49), (1.50), (1.53), (1.54) and (1.55), we get m
(r, -,1') < log+ PI + 2log+ - + log+ PI 1
r
+
~
i
T(PI'
f) + log+ log+ -I1-I Co
+ log+ n(PI) + 4log2
log+ PI + 2log+ _1_ + log _P- + 2log+ T(p, f) PI - r P - PI 1
+ 2log+ log+ ~ + 1 + 6log2 1
1
~ 4log+ P + 31og+ - - + 2log+ - + 4log+ p-r r
1 + 4log + log + ~ + 11 log 2 + 2log 3 + 1 .
Hence we have (1.43).
T(p, f)
Fix·points and Factorization of Merom orphic Functions
22
In Theorem 1.2, it is assumed 1(0) to, 00. IT the point z = 0 is a zero or pole of the function I(z), supposed non-identically equal to zero, then in the neighborhood of the point z = 0,
I(z) = c.z'
+ c.+1z·+1 +...
(c. to) .
We can apply Theorem 1.2 to the function JI(z) = z-' I(z) and get, for R, the inequality
o< r < p
0 - 8 log+ r if 8 < 0 , N(r,/d = N(r, I) - n(O, J) log r ,
m(r, z-.) =
where n(O, I) is equal to 0 or
T(r,
8
{
log + 1
if
8
r
-8, according to 8 > 0 or 8 < 0, we have
Id ~ T(r, J) + 181 (log+ r + log+ ;:)
(1.58)
(1.56), (1.57) and (1.58) yield m
(r, -I1') < 4log + T+ (p, J) + 3 log + - - + 8log + p + 6 log+ 1
1
p- r
r
1 + 4log+ log+ ~ + 5log+ 181 + 9log2 + 16,
Nevanlinna s Theory of Meromorphic Functions
23
where T+ (p, f) = max{T(p, I), o}. Evidently this inequality also holds, when 1(0) =1= 0,00. We have therefore the following corollary:
Corollary 1.1. Let I(z) be a meromorphic function non-identically equal to zero in a domain Izl < R (0 < R ~ 00). Then for 0 < r < p < R we have m
< 4log + T+ (p, f) + 3 log + - - + 8 log + P + 6log + (r, -1') 1 p-r r 1
1
1
+ 4log+ log+ ~ + 5log+ lsi + 25.
(1.59)
Sometimes we need the following generalization of Corollary 1.1:
Theorem 1.3. Let I(z) be a meromorphic function non-identically equal to zero and n 2: 1 a positive integer. Then there are positive constants A, B, C, D such that for 1 ~ r < p we have m
( rI(n)) 'j
0, we have
t m (r, f 3=1
~ a.) + m(r,J) ~ 2T(r,J) 3
Nt{r) + S(r) ,
(1.86)
29
Nevanlinna's Theory of Meromorphic Functions
where
Nd r) = {2N(r, J) - N(r, I')}
+N
(1.87)
(r, ;,)
and S (r) satisfies the following conditions: 10 IT the order of J(z) is finite, then
S(r) = O(log r) .
(1.88)
20 In the general case, there is a sequence of intervals {Ip} of finite total length and depending only on J(z), such that when r is exterior to {Ip}, we have S(r) = O{logT(r,J) +logr} . (1.89) This theorem is one form of the second fundamental theorem.
Proof. The method of proof is to introduce the auxiliary function 1
q
F(z) =
?= J(z) -
a.
1=1
1
and to find a lower bound and an upper bound of m(r, F). To find a lower bound of m(r, F), consider a value r> O. Set
0= min{l, laj - akl (1
~
i,k
~
q,i ¥: k)}
and define E j to be the set of values IP of the interval 0 the inequality
~
IP
~
21r, satisfying
. 0 IJ(re''P) - ajl < 2q .
IT z = re''P with IP E E j , then
F(z) =
1 {I + t;;."
J(z) - aj
J(z) - aj } J(z) - ak '
o 43 0
IJ(z) - akl ~ laj - akl-IJ(z) - ajl > 0 - 2q ~
" IJ(z) J(z) f;; IF(z)1 >
aj
I
2
2
ak < q 3q = 3" '
~I f(z)l_ aj
I,
(k
¥: i) ,
Fix-points and Factorization of Meromorphic Functions
30
hence log+
IF(rei~)1 ~ log+ I f( re'~. \
I-IOg3
- aj
(IP E E j )
-
Noting that the sets E j (j = 1,2, ... , q) are mutually disjoint, we have
m(r,F) ~ -1 Lq 211"
j=1
1 q ~211" " ~
1 1 E.
log+ IF(rei~)ldIP
J
E
j=1
log+
;
1
I/( re'~ .) -
d IO 3 a . I IP- g . J
Denoting by H j the complement of E j with respect to the interval 0
~
IP
~
211", then
-
11
211"
E;
log+
1
1
I(rei~)
- aj
dIP
1
=
(1) 1r-, aj
m
- -1
211"
log +1
r --
aj
1)
q ( m(r,F) ~Lm r, I-a. 3=1
H;
( , 1-1)
->m hence
1
1
I(rei~)
- aj
2q -logC '
2
-qlog cq -log3.
(1.90)
3
Now to find an upper bound of m(r,F), we write 1
F(z)
E q
= I'(z)
I'(z)
I(z) - aj
and obtain
m(r, F)
~ m (r, ;,) +
t,
m (r,
I~' aj) + log q .
Since m (r, ;,)
1dIP
= m(r, 1') + N(r, 1') - N (r, ;,) + log
I~I
Nevanlinna's Theory of Meromorphic Functions
31
by (1.16), where c =j; 0 is a constant, and
m(r, I') $ m(r, j) + m (r,
~)
,
it follows that
m(r, F) $ m(r, I) + N(r, 1') - N (r, ;,)
+m
(r, -1') + 2: m I
1') + log -I1+ log q . 1 c
(1.91)
2T(r, j) - Nl (r) + S(r) ,
(1.92)
q
(
r, _I. aJ
j=l
Inequalities (1.90) and (1.91) yield
~ m (r, I ~ aj) + m(r, j) $ where
S(r)
( f1') + ~
= m
q
r,
m
(
r,
1')
I _ a. + a
J=l
,
J
a being a constant. By Corollary 1.2 and Remark, evidently S(r) satisfies the conditions 10 and 2 0 in Theorem 1.4. Now we are going to state the second fundamental theorem in another form. For this, it is convenient to use the notations:
N(r, 00) = N(r,j),
N(r,a) = N (r, _ 1 )
I-a
(a finite)
(1.93)
introduced by Nevanlinna.
Theorem 1.5. Let I(z) be a non-constant meromorphic function and aj(j = 1,2, ... ,qj q 2: 3)q distinct values, finite or infinite. Then for r > 0, we have
q
(q - 2)T(r, j) $
2: N(r, aj) -
Ndr)
+ S(r) ,
(1.94)
j=l
where Ndr) is defined by (1.87) and S(r) satisfies the conditions 10 and 20 in Theorem 1.4.
Proof. We consider only the case that one of the values aj (j 1,2, ... ,q) is 00, for instance aq = 00. Then by Theorem 1.4, we have
I:
J=l
m (r,
I
~ a.) + m(r, j) $ 2T(r, j) J
Nl (r) + S(r) .
32
Fix-points and Factorization of Meromorphic Functions
To both sides of this inequality adding the sum Ej= 1 N (r, ai), and then noting that
T(r, f)
~T
(r, J ~
aJ
+ k i (j =
1,2, ... ,q - 1) ,
by (1.36), where ki(j = 1,2, ... ,q - 1) are positive constants, we get q
(q - 2)T(r,f) ~ LN(r,ai) - Ndr) + Sl(r) , i=l where Sl(r) = S(r) + I:j:~ ki evidently also satisfies the conditions 1° and 2° in Theorem 1.4. The case that the values ai(j = 1,2, ... ,q) are all finite, is treated by the same method. Now let us study the term Ndr) defined by (1.87). Ndr) consists of two parts 2N(r, f) - N(r, 1') and N(r, 1/1'). They are both non-negative for r ~ 1. Consider first the second part N(r, 1/1'). This part is related to the points Zo such that J(zo) is finite and Zo is a zero of order greater than one of the function J(z) - J(zo). For simplicity, let us name such a point Zo a multiple point of the first kind of J(z) and the order of Zo as a zero of J(z) - J(zo) the order of zoo Evidently n(t, 1/1') is equal to the number of multiple points of the first kind of J(z) in the disk Izl ~ t, each one of such points being counted as many times as its order minus one. Consider now the first part 2N(r, f) - N(r, 1'). This part is related to poles of order greater than one of J(z), namely multiple poles of J(z). Evidently 2n(t, f) - n(t, 1') is equal to the number of multiple poles of J(z) in the disk Izl ~ t, each multiple pole being counted as many times as its order minus one. Thus if we denote by n1 (t) the number of multiple points (those of the first kind and multiple poles) of J(z) in the disk Izl ~ t, each multiple point being counted as many times as its order minus one, then
ndt)
= {2n(t, f) - n(t, I')} + n (t, ;,)
(1.95)
Consequently we have the formula
Ndr)
=
l
o
r
n1(t) - ndO) t
dt+ndO)logr.
(1. 96)
Nevanlinna's Theory of Meromorphic Functions
33
The notation N(r, a) (a finite or infinite) introduced above may be expressed as
N(r,a) =
r n(t,a) -t n(O,a) dt+.n(O,a)logr,
10
(1.97)
where the meaning of n(t, a) is self-evident. Nevanlinna also introduced the following notation:
-
N(r, a) =
1" o
n(t, a) - n(O, a) dt + n(O, a) log r , t
(1.98)
where n(t, a) denotes the number of the roots in the disk Izl ~ t of the equation J(z) = a, each root being counted once. Noting that for any q distinct values, aj (j = 1,2, ... , q) finite or infinite, we have q
L
q
n(t, aj) - ndt) ~
j=1
L n(t, aj) , j=1
we deduce from Theorem 1.5 and (1.95) the following theorem: Theorem 1.6. Let J(z) be a non-constant meormorphic function and = 1,2, ... , qj q ~ 3)q distinct values finite or infinite. Then for r > 0 we have
ajU
q
(q - 2)T(r, 1) ~
L
N(r, aj) + S(r) ,
(1.99)
j=1 where S (r) satisfies the conditions 10 and 2 0 in Theorem 1.4. In what follows, we give some applications of the Theorems 1.4, 1.5 and 1.6. Corollary 1.3. Let J(z) be a transcendental meromorphic function. Then for each value a finite or infinite, the equation J(z) = a has an infinite number of roots, except for at most two exceptional values. This Corollary is Picard theorem for meromorphic functions. An exceptional value a, if it exists, is called a Picard exceptional value of J(z). Proof. In the particular case q = 3, Theorem 1.5 yields the inequality 3
T(r, 1) ~
L j=1
N(r, aj) + S(r) .
(1.100)
Fix-points and Factorization of Meromorphic Functions
34
Now suppose that for three values aiU = 1,2,3), the equations j(z) aiU = 1,2,3) all have at most a finite number of roots. Then evidently
N(r,ai)=O(logr)
U=1,2,3).
Consequently by the condition 2° in Theorem 1.4, when r is exterior to a sequence of intervals {Ip} of finite total length and is sufficiently large, we have T(r,j) ~ K{logT(r,f) +logr}, where K is a positive constant. But this is impossible, by (1.38). Corollary 1.4. Let j(z) be a transcendental meromorphic function of finite positive order p (0 < P < 00). Then for each value a finite or infinite, we have log n(r, a) 1-;(1.101) 1m = p, r-+oo log r except for at most two exceptional values. This Corollary is Borel's theorem for meromorphic functions. An exceptional value a, if it exists, is called a Borel exceptional value of j(z). Proof. Noting first that, for r
n(r, a) log2 ~
~
1
2r
r
1, we have
n(t a) -t-'-dt ~ N(2r, a) ,
and then by the first fundamental theorem,
n(r, a) log 2
~
T(2r, f) + k ,
where k is a positive constant. So we have
-1. log n(r, a) 1m < p. log r -
r-+oo
(1.102)
Now suppose that there are three values aiU = 1,2,3) which do not satisfy (1.101). Then we can find a constant )(0 < ) < p) such that
n(r,ai) < r'"
(r ~ ro)
Hence
N(r,ai)-N(ro,ai) =
l
r
ro
n(t,ai) dt< t
U = 1,2,3) .
l
r
ro
t'" 1 '" -ro). '" -dt=;:(r t
(1.103)
Nevanlinna's Theory of Meromorphic Functions
35
From (1.100), (1.103) and the relation S(r) = O(log r), it follows that, for sufficiently large values of r,
T(r, f) < hr>' , where h is a positive constant. But this is impossible because A < p. Consider a transcendental meromorphic function J(z) and a finite value a. By (1.36) we may write
Dividing both sides of this equality by T(r, f) and then taking lower limit, we get . m hm
r~
(r, J~a) T(r, f)
=
1-
_. N (r, J~a) hm ---'-:------,----'r-+oo
T(r, f)
(1.104)
We have also lim m(r, f) r~
=
1 _ lim N(r, f) .
T(r, f)
r-+oo
(1.105)
T(r, f)
Nevanlinna introduced the notation 6(a, f) defined for a finite or infinite as follows:
C()
- . N(r, a)
a, J = 1- r~~ T(r, f)
o
(1.106)
Evidently
o~
6(a, f) ~ 1 .
Corollary 1.5. Let J(z) be a transcendental meromorphic function and aj{j = 1,2, ... , qj q ~ 2)q distinct finite values. Then q
L 6(aj, f) + 6(00, f) ~ 2 .
(1.107)
j=l
Proof. By (1.86), we have
t - l - m (r _1_) j=l T(r, f) , J - aj
+ m(r,f) < 2+ ~ T(r, f) -
T(r, f) .
36
Fix-points and Factorization of Meromorphic Functions
Taking lower limit and making use of (1.104) and (1.105), we get
f; 6(a;, f) + q
S(~
•
6(00, f)
~ 2 + r~~ T(r, f) .
Since S(r) satisfies the condition 2° in Theorem 1.4, we have
lim~
0, then a is called a deficient value of J(z) or a Nevanlinna exceptional value of J(z), and 6(a, f) the deficiency corresponding to the value a. It is easy to see that J(z) can have at most countably many deficient values. In fact if we denote by ak the set of the deficient values satisfying the inequality 1/(k + 1) < 6(a, f) ~ 11k, and by a the set of all the deficient values, then 00
a=
Uak k=1
and, by (1.107), ak consists of at most 2(k
+ 1)
deficient values. We have
L6(a,f) ~ 2,
(1.108)
a
where the summation is taken with respect to all the deficient values of J(z). In particular for a transcendental entire function J(z), we have 6(00, f) = 1 and hence (1.109) 6(a, f) ~ 1 .
L
a;too
Now we introduce another important notion that of completely multiple value. Consider a transcendental meromorphic function J(z). A finite value a is called a completely multiple value of f(z), if each zero of J(z) - a has an order greater than one. 00 is called a completely multiple value of J(z), if each pole of f(z) has an order greater than one.
Corollary 1.6. Let J(z) be a transcendental meromorphic function. Then J(z) has at most four completely multiple values. Proof. Assume that J(z) has five completely multiple values ai(j 1,2,3,4,5). In Theorem 1.6 taking q = 5, we get 5
3T(r, f) ~
L ;=1
N(r, ail + S(r) .
=
Nevanlinna's Theory of Meromorphic Functions
Since aj is completely multiple, we see that for r -
N(r, aj) ::;
1
2N(r, aj)
1
::; 2T(r, J)
~
37
1,
+ h, (j = 1,2,3,4,5)
where h is a positive constant. Consequently 1
2T(r, j) ::; 5h + S(r) which leads to a contradiction in taking account of the condition 2° in Theorem 1.4. In particular if J(z) is a transcendental entire function, then J(z) has at most two finite completely multiple values. In fact, if J(z) has three finite completely multiple values aj(j = 1,2,3), then in Theorem 1.6, taking q = 4, a4 = 00, we get 3
2T(r, j) ::;
E N(r, aj) + S(r) j=1
which also leads to a contradiction. After more than fifty years later and by following earlier work of C. Chuang, Frank-Weissenborn and C. Osgood, N. Steinmetz have now been able to present a most convincing proof of the following generalized result which was raised by Nevanlinna in 1929.
Theorem. (Nevanlinna's second fundamental theorem for small functions). Let J(z) be a transcendental meromorphic function and a1(z), a2 (z), . .. ,aq (z) be distinct q(~ 2) meromorphic small functions (including constant function) satisfying
T(r, ai(z)) = S(r, j)
as
r
-+
00
(i = 1,2, ... ,q) .
Then, for any e > 0,
t
m (r,
Corollary.
J ~ ai) + m(r, J) ::; (2 + e)T(r, j) + S(r, j) .
E 6(a(z), j) + 6(00, j) ::; 2 , a(z)
Fix-points and Factorization of Meromorphic Functions
38
where the summation is over all deficient functions of f including constants; a(z) is called a deficient function if T(r,a(z)) = S(r, J) and 1 -
r~~N (r, f _la(z)) /T(r,J) > o.
The basic ingredient of the proof on the above result is the success in replacing the lemma of logarithmic derivative (Theorem 1.2) by an estimation of m(r, P(J)I fh), P(J) a differential polynomial of f and h a positive integer.
Rem.ark. It is natural and interesting to find some non-trivial applications of the generalized results in the studies of fix-points and factorization theory of merom orphic functions. 1.7. SYSTEMS OF MEROMORPHIC FUNCTIONS In this paragraph our main purpose is to give a complete proof of the following theorem of Borel on systems of entire functions:
Theorem. 1.1. Let fj(z)(i
(n
=
1,2, ... , n) and gj(z)(i
=
1,2, ... , n)
~ 2) be two systems of entire functions satisfying the following condi-
tions:
1) ~;'=l Ij(z)egj(z) == O. 2) For 1 :::; i :::; n, 1 :::; h, k :::; n, hi- k, the order of Ij (z) is less than the order of eg,,(Z)-gk(Z): p(Jj) < p(eg,,-gk). Then IJ(z) == 0 (i = 1,2, ... ,n). This theorem has important applications in the theory of fix points and factorization of meromorphic functions. It gives rise to the research of Nevanlinna on systems of merom orphic functions. His main result is the following theorem:
Theorem. 1.S. Let !pj(z)(i = 1,2, ... ,n) be n linearly independent meromorphic functions satisfying the identity !PI
Then for 1 :::;
i :::;
T(r, !Pj) :::;
= 1.
(1.110)
+ N(r,!pj) + N(r, D) + S(r) ,
(1.111)
+ !P2 + ... + !Pn
n, we have
t
k=l
N (r,
~) !Pk
39
Nevanlinna's Theory of Meromorphic Functions
where !Pl
!P2
!P~
!P~
!Pn
!P~ (1.112)
D= (n-l) !Pl
(n-l)
!P2
...
(n-l)
!pn
and when r is exterior to a sequence of intervals {Jp } of finite total length,
S(r) = O{logT(r)
+ logr}
,
(1.113)
where
T(r) =
m~x
l~J~n
T(r, !Pi) .
(1.114)
Proof. Differentiating (1.110) successively, we get (k) !Pl(k) +!P2
+ ... +!Pn(k)
(k - 1, 2 , ... , n - 1) .
-- 0
(1.115)
Since !Pi(j = 1,2, ... ,n) are linearly independent, D is not identically equal to zero and, by (1.110) and (1.115), we have D
=
Di
(j = 1,2, ... ,n) ,
(1.116)
where Di is the minor corresponding to !Pi in D. Hence (1.117)
(1.118)
!Pl
!P2
!Pn
and ~l is the minor corresponding to the element 1 in the first row and the first column of ~. From (1.117), we get
m(r,!pd S;
m(r'~l) + m (r, ~)
S;
m(r,~d + m(r,~) + N(r,~) + h, (1.119)
Fix-points and Factorization of Meromorphic Functions
40
where h is a constant_ Next from
D
!:l.=---IPI IP2 ••• IPn
we get
N(r,!:l.)
~ N(r, D) + t
N (r,
j=1
~)
(1.120)
IPJ
On the other hand, if we set
Sdr)
m(r, !:l.1)
=
+ m(r,!:l.) + h ,
then by (1.118) and Corollary 1.2, it is easy to see that there is a sequence of intervals {Jp } depending only on IPj(j = 1,2, ... n) and of finite total length, such that when r is exterior to {Jp }, we have
Sdr) = O{log T(r) + log r} .
(1.121)
Consequently (1.122) Similarly we have
T(r,IPj)
~
t
k=1
N
(r,~) + N(r, D) + N(r,IPj) + Sj(r) IPk
(1.123)
(j = 2,3, ... , n) , where Sj(r) is such that, when r is exterior to {Jp }, we have
Sj(r) = O{logT(r)
+ logr} (j = 2,3, ... ,n) .
(1.124)
Finally defining
S(r) = m!IX Sj(r) /$;J$;n
we get (1.111) and (1.113) from (1.122), (1.123), (1.121) and (1.124). Theorem 1.9. Let fj(z)(j = 1,2, ... , nj n 2: 2) be n merom orphic functions satisfying the following conditions:
41
Nevanlinna's Theory of Meromorphic Functions
1°
2:i=1 Cj Ij (z)
2°/j(z)
~ 0
== 0, where Cj(j = 1,2, ... ,n) are constants. (j = 1,2, ... ,n) and for 1 ~ j,k ~ n,j"# k,/j(z)/lk(z) is
not a rational function.
3° N(r, Ij) = o{r(r)}, N(r, 1/lj) = o{(r(r)}(j = 1,2, ... ,n), where r(r) = n:tin I~"k~n jt:-k Then
Cj
=
0 (j
=
Jk
1,2, ... , n).
Proof. Consider first the case n
Assume that Cj(j then
{T (r, ~j) }
=
=
2. Then
1,2) are not both equal to zero, for example
C1
"#
0,
h(z) _ Cz h(z) =-~
which is incompatible with condition 2°. Consequently Theorem 1.9 holds when n = 2. Now assume that Theorem 1.9 holds for an integer n ~ 2, and let us show that it is also true for n + 1. In fact, consider n + 1 meromorphic functions Ij(z)(j = 1,2, ... ,n+l) satisfying the conditions in Theorem 1.9, so that n+1
L
cj/j(z) == 0 .
(1.125)
j=1
Suppose that Cj(j = 1,2, ... ,n + 1) are not all equal to zero. Then Cj(j = 1,2, ... ,n + 1) must be all different from zero. In fact, if for example Cn +1 = 0, then n
L cj/j(z) == 0 j=1
and Ij(z)(j = 1,2, ... ,n) satisfy the conditions in Theorem 1.9. Since by assumption Theorem 1.9 holds for the integer n, we have Cj = 0 (j = 1,2, ... ,n) and hence Cj = 0 (j = 1,2, ... ,n + 1), contrary to the hypothesis that Cj (j = 1,2, ... ,n + 1) are not all equa1 to zero. So Cj "# 0 (j = 1,2, ... ,n + 1). Set
lr>,'(Z) =_ r
c;lj(z)( )
cn +1/n+1
Z
(J, = 1,2, ... ,n ) .
(1.126)
42
Fix-points and Factorization of Meromorphic Functions
By (1.125)' we have n
L r A ' • It follows that (1.135) and (1.136) are also true. So condition 3) in Theorem 1.10 is also satisfied. Theorem 1.7 is now completely proved.
Corollary 1.'1. Let fi(z)(i = 1,2, ... ,n+l) and gi(z)(j = 1,2, ... ,n) (n ~ 1) be two systems of entire functions satisfying the following conditions:
Ei=l
fi(z)eUj(z) == fn+1(z). 2) For 1 ~ j ~ n+ 1, 1 ~ h ~ n, the order of fi(z) is less than the order of eUA(z). In case n ~ 2, for 1 ~ j ~ n + 1, 1 ~ h, k ~ n, h ¥= k, the order of fi(z) is less than the order of eU"(Z)-Uk(Z).
1)
Then fi(z)
== 0 (i = 1,2, ... ,n+ 1).
Proof. It is sufficient to note that, in setting gn+dz) = 0, we have n
L
fi(z)eUj(z) - fn+dz)eu,,+.(z)
i=l
and then Corollary 1. 7 follows from Theorem 1. 7.
== 0
,
2 FIX-POINTS OF MEROMORPHIC FUNCTIONS
2.1. INTRODUCTION In the theory of fix-points of meromorphic functions, Nevanlinna's theory of meromorphic functions and Montel's theory of normal families play an important role. So in this chapter we first give some complements of Nevanlinna's theory of meromorphic functions sketched out in the previous chapter and prove some theorems on fix-points due to Rosenbloom and Baker. Next we give an account of the main points of Montel's theory of normal families of holomorphic functions and apply it to Fatou's theory of fix-points of entire functions. 2.2. SOME THEOREMS ON MEROMORPHIC FUNCTIONS We first prove the following theorem which is a generalization of Nevanlinna's second fundamental theorem.
Theorem 2.1. Let I(z) be a non-constant meromorphic function. Let = 1,2,3) be three distinct meromorphic functions such that
'2 is a constant. Now it is evident th~t S(r) also satisfies the conditions 10 and 2° in Theorem 2.1. Similarly we can prove the following theorem:
Theorem 2.2. Let I(z) be a non-constant meromorphic function. Let !Pi (z)(i = 1,2) be two distinct meromorphic functions such that
T(r, !Pi)
= o{T(r, In
(j
= 1,2)
.
Then for r > 1 we have the inequality
T(r, f)
~ N(r, f) +
t.
~ !Pi) + o{T(r, In + S(r)
N (r, I
where S(r) satisfies the conditions 1° and 2° in Theorem 2.1. Now we are going to prove some theorems on the growth of composite functions. For this we need some preliminary lemmas and theorems.
Lemma 2.1. Let a and b be two positive numbers such that b ~ 8a 2 • Then for x ~ 2, we have ~
+ 8b
e 4 x > 8a log x
(2.10)
.
Proof. Consider the auxiliary function b.
!p(x) = e x - 8a log x - 8b . 4
Evidently it is sufficient to show that
!p(2) > 0,
!p' (x) = e
8a
b. 4
-
-
x
> 0 (x
~ 2) .
(2.11)
We have 2e!
>
2(1 +
!!.a
+ .!:. b22 ) >
1+ ( 1+
~)
2
(2.12)
2a
b a
- >
8a,
> 8a ( 1 +
~)
> 8alog2 + 8b
= 8a
+ 8b
(2.13)
Fix'points of Meromorphic Functions
53
and hence ~
2e .. > 8alog2
+ 8b
which shows that the first inequality in (2.11) holds. On the other hand, (2.12) and (2.13) imply ~ 2e .. > 8a , hence the second inequality in (2.11) also holds.
Lemma 2.2. Let U(r) be a non-negative and non-decreasing function in an interval 0 < r < p. Let a and b be two positive numbers such that b ~ 2a and b ~ 8a 2 • Assume that the inequality
R
U(r) < alog+ U(R) + alog R _ r + b
(2.14)
holds for 0 < r < R < p. Then the inequality
R
U(r) < 2a log R _ r + 2b holds for 0 < r < R < p. This Lemma in a different form was obtained by Borel in his fundamental paper "Sur les zeros des fonctions entieres", Acta Math. 20 (1897). It is put in the present form by Bureau and Milloux.
Proof. Let rand R be two values such that 0 < r < R < p and assume
R
U(r) ~ 2alog R-r +2b.
(2.15)
We are going to show that the two values r' = (r + R)/2 and R also satisfy the inequality
U(r')
~ 2alog R ~ r' + 2b .
In fact, by (2.14)' we have
U(r)
- e " - - . 4 R-r'
-eo. - -
On the other hand, by Lemma 2.1, taking x have k
(2.17)
= RI(R - r') in (2.10), we
R R > 8a log - - + 8b . R-r' R-r'
(2.18)
eo. - -
Inequalities (2.17) and (2.18) yield (2.16). Now let rn
=
(rn-l
+ R)/2 (n = 1,2, ... ),
ro
=
r.
Then for each n, we have
0< rn < R < P and
R U(rn) ~ 2alog R _ rn
(2.19)
+ 2b .
(2.20)
U(r) being a non-decreasing function, (2.19) implies U(rn) ~ U(R). On the other hand, (2.20) implies U(rn) -+ 00, as n -+ 00. So we get a contradiction. Lemma 2.3. Let a > e and x > 0 be two numbers. Then we have 1 logx + alog+ log+ - ~ a(loga - 1) x
+ log+ x
.
(2.21)
Proof. If x > lie, then log+ 10g+(1lx) = 0, hence (2.21) holds. If (2.21) becomes
x ~ lie, then
log x
+ a log log -1 x
~
a(log a - 1) .
(2.22)
55
Fix-points of Meromorphic Functions
Consider the function
rp(y) = alogy - y - a(loga -1)
(y > 0)
(2.23)
and its derivative
rp'(y)
=
~
- 1 .
(2.24)
Y
We can see easily
rp(y)
:::=;
rp(a) = 0 (y > 0) .
(2.25)
Replacing in (2.25) y by log(l/x), we get (2.22).
Theorem 2.3. Let I(z) be a holomorphic function in the circle Izl < 1 in which I(z) does not take the values 0 and 1. Then for Izl < 1, we have log I/(z)1
0, we have (2.35) This theorem is due to P6lya, G. ("On an integral function of an integral function", J. London Math. Soc., 1 (1926) 12-15).
Proof. Consider the function
which evidently satisfies the conditions of Theorem 2.4. There is then a circle Iwl = R, R ~ c (0 < c < 1), such that in the circle IZI < 1, the function H(Z) takes every value w of the circle Iwl = R. Hence in the circle Izl < r, the function h(z) takes every value W of the circle IWI = RM (~,h). Set
61
Fix-points of Meromorphic Functions
Let Wo be a point of the circle IWI = R', such that Ig(Wo)1 = M(R',g) and let Zo be a point of the circle Izl < r, such that h(zo) = Woo Then we have
M(r,f)
~
I/(zo)1 =
Ig{h(zo)} I = Ig(Wo)1
= M(R',g) ~ M {cM G,h) ,g} Corollary 2.1. Let g(z) and h(z) be two entire functions and let I(z) = g{h(z)}. IT h(z) is non-constant, there is a number CI (0 < CI < 1) such that when r is sufficiently large, we have
M(r,f) ~ M {cIM G,h) ,g} .
(2.36)
Proof. Let gdz) = g(z + h(O)), hdz) = h(z) - h(O). Then I(z) gl {hdz)} and by Theorem 2.5,
M(r,f) ~ M {cM
G,hl) ,gd
=
.
Since
g(z) = gdz - h(O)),
h(z) = hl(z) + h(O) ,
we have
M(r, g)
~
M(r + Ih(O) I, gl),
M(r, h)
~
M(r, hd + Ih(O) I .
Hence when r is sufficiently large, we have
M(r,g) and
M (r, f)
~
M(2r,gd,
M(r,h) < 2M(r, hI)
~ M { ~ M ( ~, h) ,gl } ~ M { ~ M (~ , h) , g}
Now we prove another theorem of P6lya, also based upon Theorem 2.4.
Theorem 2.6. Let I(z) and g(z) be two non-constant entire functions such that the order of the function 0 and c = 1(0). Then the function
F(Z) =
I (!rZ) - c MUr,1 - c)
Fix-points and Factorization of Meromorphic Functions
62
satisfies the condition of Theorem 2.4_ Hence there is a circle Iwl = R (R > A) such that in the circle IZI < 1 the function F(Z) takes every value w of the circle Iwl = R_ It follows that in the circle Izl < r/2 the function I(z) takes every value W of the circle IW - e I = RM (~, I - e) _ Set
el = R' such
and let Wo be a point of the circle IW -
Ig(Wo)1 and let
Zo
=
be a point of the circle
max
IW-cl=R'
Izl
ro, where ro > 1. Hence
T(r N , g) - T(ro, g) > T(r, g) - T(ro, g) N log r - log ro
-
log r - log ro
and, when r is sufficiently large, we have
N
T(r N , g) ~ 2{T(r, g) - T(ro, g)} > and
N
"4 T (r, g)
,
N
T(r, ip) > 12 T(r, g) . Since N is arbitrary, we have (2.38). Theorem 2.B. Let J(z) be a transcendental entire function and g(z) a transcendental meromorphic function. Let ip(z) = g{J(z)}. Then lim T(r, ip) r--+oo
T(r, J)
= 00
.
(2.39)
64
Fix-points and Factorization of Merom orphic Functions
Proof. Assume first that g(z) has an infinite number of zeros. Choose p zeros ai(i = 1,2, ... ,p) of g(z), such that
lai - ajl > 1
(i,j = 1,2, ... ,pji"l j) .
:5
Then it is easily seen that we can find a number 0 < 8 1/2 and a number M > 0 such that in the p circles Ci : Iz - ail < 8(i = 1,2, _.. , p) we have respectively the p inequalities
Ig(z)1 < Mlz-ail
(i= 1,2,._. ,p).
Evidently any point z belongs at most to one of the circles Ci (i 1,2, ... ,pl. Consequently p
log
+1",
Ig(z)1 ~
8 log+ Iz _8 ail -log + 8M, p
log and m
+
1
1~(z)1 ~
8 log + I'(z)8_ ail-log + 8M, '"
(r,;;) ~ ~ m (r, ,~aJ -log+ 8M .
Then making use of the inequality m
(r, -, 1 ):5 m (r, _, 8 ) + log! , -at -at 8
we get (2.40) where h is a constant. On the other hand, we see easily that (2.41) Inequalities (2.40) and (2.41) yield
65
Fix-points of Meromorphic Functions
Finally making use of the relations
T (r, f
~
aJ
= T(r, f)
+ 0(1), T (r, ~)
= T(r, IP)
+A,
we get
T(r, IP) ;::: pT(r, f)
+ 0(1)
,
and when r is sufficiently large,
T(r, IP) T(r,f) >
p
2.
Since p is arbitrary, we have (2.39). In the above we have assumed that g(z) has an infinite number of zeros. In general, since g(z) is a transcendental meromorphic function, there is a finite value Wo such that the function g1(Z) = g(z) - Wo has an infinite number of zeros. Set
IPdz)
= g1 {f(z)} = g{f(z)} -
Wo
= IP(z) -
Wo .
By the result just obtained, we have
lim T(r,IPd = r-oo
00 •
T(r, f)
Then by the inequality
T(r,IPd ~ T(r, IP)
+ log+ Iwol + log 2 ,
we get again (2.39). Note that in the proof of Theorem 2.8, the condition that f(z) is a transcendental entire function is not necessary. What is important is that f(z) is a non-constant entire function. 2.3. SOME THEOREMS OF ROSENBLOOM ON FIX-POINTS For reference, see "The fix points of entire functions", Medd. Lunds Univ. Mat. Sern., Suppl.-Bd. M. Riesz 186 (1952). Definition 2.1. Let f(z) be a meromorphic function. A point zo(zo too) is said to be a fix-point of f(z), if f(zo) = zoo This is equivalent to say that Zo is a zero of the function f (z) - z. Theorem 2.9. Let P(z) be a polynomial of degree n ;::: 2 and let f(z) be a transcendental entire function. Then the function P{f(z)} has an infinite number of fix-points. Proof. Suppose, on the contrary, that the function P{f(z)} has at most a finite number of fix points. Then the function P{f(z)} - z has at
Fix-points and Factorization of Meromorphic Functions
66
most a finite number of zeros, hence
P{f(z)} - z = Q(z)ea(z)
(2.42)
where Q(z) t 0 is a polynomial and o:(z) is an entire function_ Since the left hand member of (2.42) is a transcendental entire function, so o:(z) is non-constant. The equation f{P(z)} = z has necessarily at most a finite number of solutions. In fact, if Zo is a solution of this equation, then
f{P(zo)} = Zo , hence
P(f{P(zo)}) = P(zo) and by (2.42),
Q{P(zo)} = 0 . So Zo is a zero of the polynomail Q{P(z)}. Since Q{P(z)} most a finite number of zeros. By Theorem 2.8, lim T(r, f(P)) r-+oo
hence
T(r, P)
= 00
t
0, it has at
,
f {P(z)} is a transcendental entire function and we have f{P(z)} = L(z)eP(z)
+z
(2.43)
where L(z) to is a polynomial and ,B(z) is a non-constant entire function. From (2.42) and (2.43) we get
P{L(z)eP(z)
+ z} = P(f{P(z)}) = Q{P(z)}ea{P(z)} + P(z) .
(2.44)
Let
P( Z)
= CoZ n + C1Zn-l + ... + Cn (co i' 0,
n? 2) .
After calculation, we get n
L i=l
ui(z)eiP(z)
+ v(z)ea{P(z)}
= 0
(2.45)
Fix-points of Meromorphic Functions
67
where uj(z)(i = 1,2, ... , n) and v(z) = -Q{P(z)} ~ 0 are polynomials. It is easy to see that the polynomial Uj(z) and the polynomial (2.46) have the same degree and have the same coefficient for the term of the highest degree. Consequently the polynomials Uj(z) U = 1,2, ... , n) and v(z) are all non-identically equal to zero. N ow distinguish two cases: 1) The functions J'{3(z) - a{P(z)}(i = 1,2, ... , n) are all non-constant. In this case, by (2.45) and Theorem 1.7, uj(z)(i = 1,2, ... , n) and v(z) are all identically equal to zero. So we get a contradiction. 2) There is an integer io (1 ~ io ~ n) such that io/3(z) - a{P(z)} is a constant. In this case, for i =I- J'o (1 ~ i ~ n)'J'/3(z) - a{P(z)} is nonconstant, for otherwise, /3(z) would be a constant. Writing (2.45) in the form n
L
I
uj(z)ej.B(z) + vdz)eo{P(z)} = 0
j=l
2:'
the term ui(z)eio.B(z) is omitted. Since n ~ 2 and i/3(z) - a{P(z)}(1 ~ J- ~ n,i =I- J'o) are non-constant, we get again a contradiction by Theorem 1.7. where in the sum
Lemma 2.4. Let n. If Zo is a root of the equation
then and hence Z = fm-n(zo) is a fix-point of fn(z). It follows that, either for a certain j(1 ~ j ~ p) we have
or Z is a fix-point of exact order no(1 ~ no ~ n - 1) of f(z), in the latter case, we have
fno(Z)
=Z ,
that is
fm-n+no (zo) = fm-n(zo) . Summing up these results, we conclude that Zo is a zero of one of the following functions:
fm-n(z) -
Zj
(j = 1,2, ... ,p) ,
fm-n+no (z) - fm-n(z)
(no
=
1,2, ... ,n - 1)
and hence
N
(r, fm - 1fm-n )
t
~ j=1 N + = 0
(r, fm-n1 -
) Zj
~-( 1 - - -) ~ N r, - - - -
no=1
fm-n+no - fm-n
{~ T(r, f.) } = o{T(r, fm)} .
(2.51)
72
Fix-points and Factorization of Merom orphic Functions
Now applying Theorem 2.2, with Im(z) for I and Im-n(z), Z for If'i(i = 1,2), we get, for r > 1,
T(r, 1m) :5 N (r, 1m _l/m_n) + N (r, 1m 1_
Z) + o{T(r, 1m)} + S(r)
where S(r) satisfies the conditions 10 and 2 0 in Theorem 2.1 with respect to Im(z). Consequently when r is exterior to a sequence of intervals of finite total length, we have
T(r, 1m) :5 N (r, 1m 1_
Z) + o{T(r, 1m)} .
(2.52)
Now divide the zeros of Im(z) - Z into two kinds: 1) A zero Zo of 1m (z) - Z belongs to the first kind, if Zo is not a zero of one of the functions f,,,(z) - z (k = 1,2, ... ,m - 1), in other words, Zo is a fix-point of exact order m of I(z). 2) A zero Zo of Im(z) - z belongs to the second kind, if Zo is a zero of one of the functions Ik (z) - z (k = 1,2, ... ,m - 1). By (2.52) and the relation
~ N (r, Ik ~ z) = 0 {~T(r'lk)} = o{T(r, 1m)} , evidently Im(z) - z has an infinite number of zeros of the first kind, hence I(z) has an infinite number of fix-points of exact order m. So far we have proved that if there is a positive integer n such that I(z) has at most a finite number of fix-points of exact order n, then for any positive integer m > n,/(z) has an infinite number of fix-points of exact order m. The conclusion of Theorem 2.12 is therefore valid.
Theorem 2.13. Let I(z) be a polynomial of degree d ~ 2. Then for each positive integer n,/(z) has at least one fix-point of exact order n, except at most one positive integer n. Proof. Evidently the degree of In (z) is d n and hence for each positive integer n,/(z) has fix-point of order n. In particular, I(z) has fix-point of order 1, and hence fix-point of exact order 1. Suppose, on the contrary, there exist two positive integers nand k such that n > k ~ 2 and that I(z) has neither fix-point of exact order n, nor fix-point of exact order k. Consider the rational function
( ) = In(z) - z In-k(Z) - z
If' z
Fix-points of Meromorphic Functions
73
Denoting by No the number of the zeros of !p(z), each zero being counted only once, we are going to prove that we have (2_53) In fact, if Zo is a zero of !p(z), then Zo is a zero of fn(z) - z_ Since by assumption, f(z) has no fix-point of exact order n, Zo is a fix-point of exact order ;(1 :5 ; < n) of f(z). n must be divisible by;. In fact, this is clear, if; = 1. IT 1 4. In this case, first note that n is not divisible by n - 1 and n - 2. In fact, if n is divisible by n - 1, then n = m(n - 1) (m is a positive integer with 1 < m < n), and therefore (m - l)n = m which is obviously impossible. Similary, ifnis divisible by n-2, then n = m(n-2)(1 < m < n) and therefore (m - l)n = 2m, 2n = 2m(n - 2) = (m - l)n(n - 2),2 = (m - l)(n - 2) which is impossible. Hence;:5 n - 3 and each zero of !p(z)
74
Fix-points and Factorization of Meromorphic Functions
must be a fix-point of one of the polynomials fj(z)(j = 1,2, ... ,n - 3). Consequently
n-3 N o_~ < " dj j=1
=
dn-2 _ d < dn - 2 - d d-1-
< dn - 2 .
(2.53) therefore always holds. Now denoting by N 1 the number of zeros of cp(z) - 1, each zero being counted only once, we are going to prove that (2.54) In fact, if Zo is a zero of cp(z) - 1, then
which shows that Z = fn-k(ZO) is a fix-point of ik(z). Since, by assumption, f(z) has no fix-point of exact order k, hence Z is a fix-point of exact order j with 1 :=; j < k. As in the above, we see that k is divisible by j. Distinguish two cases: 1) k ~ 3. In this case, we see, as in the above, that k is not divisible by k - 1, hence 1 :=; j :=; k - 2. Since
fj(Z) = Z,
Z = fn-dzo)
shows that Zo is a zero of the polynomial fn-k+j(Z) - fn-k(Z), we have
k-2 N1 :=;
L
j=1
n-2 dn-k+j
1 of cp(z) or cp(z) - 1, is a zero of order m - 1 of cpl(Z). Hence, by (2.53) and (2.54), (2.55) On the other hand, cp(z) has a pole of order dn - dn - k at infinity. Let p be the number of the other poles of cp(z), each pole is counted according to its order of multiplicity. Then the number P of the poles of cp(z) (with due count of order of multiplicity) in the extended complex plane is given by (2.56) Evidently the number of the poles of cp(z) - 1 in the extended complex plane is also P. For cpl(Z), it has a pole of order d n - dn - k - 1 at infinity and the number of its other poles (with due count of order of multiplicity) does not exceed 2p, hence in the extended complex plane, the number pI of the poles of cp' (z) satisfies the inequality pI ~ d n
_
dn -
k -
1 + 2p .
(2.57)
Now it is well known that in the extended complex plane, the number of the zeros and the number of the poles of a rational function are equal (with due count of order of multiplicity), hence from (2.56) we have
On the other hand, from (2.57) we have N'
=
pI ~ d n
-
dn -
1 + 2p .
k -
Finally from (2.55) we get 2(dn
_
dn -
k
+ p)
~
dn _ dn ~
dn - 2
+
dn - k
+
+ dn - 1 + N ' dn - 2 + d n - 1 + dn _
~ dn - 2
dn - k ~ dn - 1 _
dn - 2
1
~ 2d n - 2
~ dn - 1
and arrive at a contradiction.
+
dn - 1 -
+
+
k -
1 + 2p ,
1 -
1 ~ dn
1,
dn - 1 -
dn - 1 _
dn -
1
1 = 2d n -
-
1 ,
76
Fix-points and Factorization of Merom orphic Functions
Consider for example the entire function I{z) = e% + z which has no fix-point, hence the positive integer n = 1, is exceptional in the sense of Theorem 2.12. On the other hand, consider the polynomial f(z) = z2 - z. We have h{z) - z = z3{z - 2), hence h{z) has two fix-points z = 0 and z = 2. Since these two points are also fix-points of I{z), hence the positive integer n = 2 is exceptional in the sense of Theorem 2.13. An interesting question is that, in the case of transcendental entire functions, whether the exceptional positive integer n in Theorem 2.12 may be different from 1. 2.5. NORMAL FAMILIES OF HOLOMORPHIC FUNCTIONS In the theory of normal families of holomorphic functions, the following theorem of Montel is fundamental.
Theorem 2.14. Let In {z)(n = 1,2, ... ) be a sequence of holomorphic functions in a domain D. IT this sequence of functions is locally uniformly bounded in D, then we can extract from this sequence of functions a subsequence In" (z)(k = 1,2, ... ) which converges locally uniformly in D. To say that the sequence In{z)(n = 1,2, ... ) is locally uniformly bounded in D means that for each point Zo in D, there is a circle c : Iz - Zo I < r belonging to D and a positive number M such that
I/n{z)l $
for n ~ 1, z E c .
M
(2.58)
Similarly we define the notion of local uniform convergence in D. For the proof of this theorem, we need two lemmas.
Lemma 2.6. Let In{z)(n = 1,2, ... ) be a sequence of functions holomorphic in a circle Izl $ R. IT the sequence In{z)(n = 1,2, ... ) is uniformly bounded for Izl $ R and converges at each point of a set 8 for which the point z = 0 is a point of accumulation, then the sequence In {z)(n = 1,2, ... ) converges uniformly in any circle Izl $ r (O < r < R). To say that a function is holomorphic in a circle Izl $ R means that it is holomorphic in a circle Izl < p with p> R. Proof. Let
00
In{z)
=
Lain) zk . k=O
Since the sequence In{z)(n = 1,2, ... ) is uniformly bounded for we have
I/n{z) -
In{O) I $
I/n{z)1 + I/n{O)1 $
2M for n ~ 1,
Izl
~
Izl R
$ R,
Fix-points of Meromorphic Functions
77
where M > 0 is a constant independent of nand z. Hence by Schwarz lemma, we have
Consider a point Zo of the circle
JzJ :5 R.
We have
Given arbitarily a number e > 0, let Zo be a point of the set
8
such that
and next let N be a positive integer such that
Jfn(zo) - fm(zo)J
1 such that in the domain Izl > R, the sequence IPn(z)l(n = 1,2, ... ) converges uniformly to 00. Proof. We have Since d ~ 2, we can find a number R > 1 such that for Izl > R, we have
Then for Izl > R, we have
IP(z)1 > 21z1 > 2R > R , IP2(z)1 > 2IP(z)1 > 221z1 > 22R > R, IP3(Z)1 > 2IP2(z)1 > 231z1 > 23R > R , in general
Fix-points and Factorization of Meromorphic Functions
102
Hence in the domain Izl > R, the sequence IPn(z) I (n = 1,2, ... ) converges uniformly to 00. Theorem 2.29 shows that the point z = 00 is one at which the family {Pn(z)(n = 1,2, ... )} is normal. On the other hand, the point z = 00 is a fix-point of P(z) and it is attractive in the sense that the point w = 0 is an attractive fix-point of the function
wd
1
p(t) =
aowd+alwd-l+ ... +ad_lw+ad
Finally we are going to prove the following theorem:
Theorem 2.30. The set J(P) has the following properties: 10 J(P) is a non-empty bounded perfect set_ 2 0 Let En be the set of the fix-points of Pn(z) and define 00
E(P) =
U En . n=l
Then each point of J(P) is a point of accumulation of E(P).
Proof. First we prove that J(P) is non-empty. In fact, if J(P) is empty, the family {Pn(z)(n = 1,2, ... )} is normal in the complex plane. Let Zo be a fix-point of P(z). We have
Pn (zo) = Zo
(n = 1, 2, ... ) .
So the sequence Pn(zo)(n = 1,2, ... ) is bounded. Then by Lemma 2.9, the sequence Pn(z)(n = 1,2, ... ) is locally uniformly bounded in the complex plane. This contradicts Theorem 2_29_ By Theorem 2.29, the set J(P) belongs to the circle Izl ~ R and therefore is bounded. To prove that J(P) is perfect, it is sufficient to show that J(P) c {J(P)}'. First of all, by Theorem 2.13, there exist three positive integers n. (i = 1,2,3) such that nl < n2 < n3 and that there are three points ~.(i = 1,2,3) which are fix points of P(z) of exact order n.(i = 1,2,3) respectively. Obviously ~.(i = 1,2,3) are distinct. Since Pnj(~.) = ~.(i = 1,2,3), we see that the sequences Pn(~.)(n=I,2,
are bounded.
... ) (i=I,2,3)
(2.90)
103
Fix-points of Merom orph ic Functions
Now consider a point Zo E J(P) and a circle c : going to show that in c there is a point z' such that
z' i= Zo, z' E
Iz - zol
'1 < o. In view of the condition a) and the boundedness of the sequences (2.90), we see that the family {Pn(z)(n = 1,2, ... )} is not normal in 0 •. Hence there are two points (\(.(i = 1,2) such that (\(. E 0. (i = 1,2), (\(. E J(P) (i = 1,2) . (2.94) Let H be a positive integer such that
h~ H .
Pm" (zo) E"f for
(2.95)
As in the proof of the part 2° in Theorem 2.24, we see that we can get an integer h ~ H such that the values taken by the function Pm" (z) in the circle c cover one of the domains 0. (i = 1, 2), say 0 1 . So in c there is a point z' such that (2.96) Then by (2.94) (\(1
E
J(Pm ,,),
z'
E
J(Pm ,,) = J(P)
and by (2.93)' (2.95)' (2.96)' we have z'
I- Zo
.
Hence the point z' satisfies the condition (2.91). 2) >. is infinite. Let H be a positive integer such that
IPm,,(zo)1
> M for h
~ H
(2.97)
where M is the number in (2.92). As above, we can get an integer h ~ H such that the values taken by the function Pm" (z) in the circle c cover one of the domains O.(i = 1,2). By (2.97), again we see that in c there is a point z' satisfying the condition (2.91). It remains to prove the part 2° of Theorem 2.30. Let ai(j = 1,2, ... , q) be all the distinct roots of the equation P'(z) = D. Consider a point Zo E J(P). Since Zo is a point of accumulation of J(P), for any circle c : Iz-zo I < r, there is a point z' E c such that
z'
E
J(P), z' I-
Zo,
z' I- P(ai) (j = 1,2, ... ,q) .
Fix-points of Meromorphic Functions
Let a be a root of the equation P(z)
P(a) = z',
= z'.
105
Then
P'(a) =I 0 .
Next as in the proof of the part 3° of Theorem 2.24, we see that there is a point Zl E c such that Zl
=I Zo,
Zl
E
E(P) .
Hence Zo is a point of accumulation of E(P). For further study of the Julia sets of polynomials and rational functions, the reader is referred to the following works: 1. H. Brolin, "Invariant sets under iteration of rational functions", Ark. Mat. 6 (1965) 103-144. 2. P. Blanchard, "Complex analytic dynamics on the Riemann sphere" , Bull. American Math. Soc. 11 (1984) 85-141.
3 FACTORIZATION OF MEROMORPHIC FUNCTIONS
3.1. INTRODUCTION The factorization theory of meromorphic functions concerns whether a given function can be expressed as a composition of two or more nonlinear meromorphic functions. This theory was developed about two decades ago. The investigation is closely related to the study of the fix-points of a function. A complex number Zo is said to be a fix-point of J{z) iff J{zo) = zoo As far back as 1926, Fatou claimed that for any nonlinear entire function J, the iteration J(f)( = h) has at least one fix-point. This fact was formally proved in 1952 by P.C. Rosenbloom utilizing Picard's theorem. The proof assumes if J(f) has no fix-point at all, then clearly, it is not possible for J(z) to have any fix-point either. Therefore, the function
J{z) - z F{z) = J(f(z)) - J{z) is entire and assumes neither 0 nor 1. According to Picard's theorem F must be a constant, say c. Clearly c =1= 1. From the above equation, we get
c[J(f(z)) - z] = J{z) - z . By differentiating both sides, we have
c[J'(f{z))j'(z) - 1] = j'(z) - 1 107
108
Fix-points and Factorization of Merom orphic Functions
or
!,(z)[c!,(f(z)) - 1] = c - 1 # 0 . It follows that J'(z) never vanishes, and moreover, J'(z) never takes the value ~. Thus J' has to be a constant, hence J is linear. This contradicts the assumption and therefore proves the assertion that J(f) must have infinitely many fix-points for any nonlinear entire function f. In the same paper, Rosenbloom extended this result and obtained two theorems (given below) by applying the newly developed Nevanlinna's valuedistribution theory. Since then, the value-distribution theory has greatly affected the research in the factorization theory.
Theorem 3.1. Let J and g be two transcendental entire functions. Then either J or J(g) must have infinitely many fix-points. Theorem 3.2. Let P(z) be a nonlinear polynomial and J(z) be a transcendental entire function. Then P(f(z)) must have infinitely many fix-points. It was in the same paper that Rosenbloom first introduced the concept of "prime function". He defined an entire function F(z) to be prime if every factorization of the form F(z) = J(g(z))(= J 0 g(z)). J, g being entire implies that one of the functions, J or g, must be linear. Rosenbloom asserted, without giving a proof, that eZ + z is a prime function and remarked that its proof was quite complicated. Given the present techniques in the study of factorization, the proof that eZ + z is prime is a relatively simple matter. It was not until 1968 that F. Gross gave a broader definition of the factorization for meromorphic functions. He not only provided a proof of the primeness of eZ + z but also started a series studies on factorization theory. In this book, the emphasis will be on the development of the methods for testing whether a given meromorphic function can be factorized as two or more nonbilinear meromorphic functions. More specifically, we shall discuss (i) the forms of the factors in a factorization; (ii) the existence of fix-points and the factorization; (iii) the criteria for pseudo-prime functions; (iv) the growth rates of meromorphic solutions of certain functional equations and (v) the factorizations of meromorphic solutions of linear differential equations and the uniquely factorizability of certain functions.
Factorization of Meromorphic Functions
109
Through the investigations of Gross and Yang in the U.S.A., Goldberg and Prokopovich in the U.S.S.R., Baker and Goldstein in England, Steinmetz in Germany, and Ozawa, Urabe and Noda in Japanj the theory of factorization has become a new branch in the value-distribution theory of meromorphic function. For the interest of the reader, we have included many open questions and research problems for further studies in this book. 3.2. BASIC CONCEPTS AND DEFINITIONS
Definition 3.1. Let F(z) be a meromorphic function. If F(z) can be expressed as (3.1) F(z) = f(g(z))(= f 0 g(z)) , where f is meromorphic and 9 is entire (g may be meromorphic when f is rational)' then we call expression (3.1) a factorization of F (or simply a factorization), and f and 9 are called the left and right factors of F, respectively.
Definition 3.2. If every factorization of F of the above form, implies that either f or 9 is bilinear (J is rational or 9 is a polynomial), then F is called prime (pseudo-prime). Definition 3.3. If every factorization of the form (3.1) leads to the conclusion that f must be a bilinear form when 9 is transcendental (or f is transcendental and 9 must be linear) then F is called left-prime (rightprime). When factors are restricted to entire functions, it is called a factorization in entire sense. Under such a provision a prime (pseudo-) function will be denoted as E-prime (E-pseudo-prime). Nevertheless, we shall prove in the sequel that if a nonperiodic entire function F is E-prime (E-pseudoprime) then it also must be prime (pseudo-prime). In other words, we only need to consider entire factors for the primeness (or pseudo-primeness) of a non periodic entire function. Until recently the majority of the research accomplishments in the factorization theory have been based on the studies of the prototype ele + Zj the construction of certain families of prime or pseudo-prime functionsj the finding of sufficient conditions for a certain class of functions being prime or pseudo-prime, and the discussions of problems of the uniquely factorizability of certain functions as well as the commutativity of factors. In all these investigations, the Nevanlinna value-distribution theory has been
110
Fix-points and Factonzation of Merom orphic Functions
used as the primary tool. Also, in the development of the proofs, the following properties of meromorphic functions have been used: (i) the growth property; (ii) the distribution of the zeros or the existence of defect values; (iii) the periodicity; (iv) the fix-points and (v) being a solution of a linear differential equation. Generally speaking, the research in the factorization theory is still in its infancy stage. There are many interesting questions to be studied and resolved. We strongly believe that value-distribution theory can be further perfected by studying the factorization theory. Here we shall only deal with factorization of transcendental meromorphic functions, since Ritt has obtained a complete theory on the factorization of polynomials (but not rational functions!). 3.3. FACTORIZATION OF CERTAIN FUNCTIONS In this section we shall prove that for any non-constant polynomial P(z)e Z + P(z) is prime; a generalized form of eZ + z. Prior to this proof it is natural for us to ask: For a given function, is there any link between the forms of the factors and that of the given function? More precisely, we ask whether certain classes of entire functions and their factors possess more or less similar properties? The answer is "yes". Before proceeding further, we introduce some definitions and lemmas below:
Definition 3.4. We shall call an entire function F(z) periodic mod 9 with period T, if and only if the following identity holds:
F(z + T) - F(z) = g(z) . Sometimes, we simply call such a function a pseudo-periodic function mod g. For instance given the function F(z) == eZ + P(z), F is periodic mod a polynomial with period 21ri.
Theorem 3.3. Let F(z) be an entire function and periodic mod a polynomial P(z) with period T. If F = fog, then 9 must assume the following form: (3.2) where Hi(Z),i = 1,2, are periodic functions with the same period constant, and q(z) is a polynomial.
T,
c is a
111
Factorizatzon of Meromorphic Functions
Proof. We may assume, without loss of generality (w.l.o.g) that According to the above hypothesis we have
T
=
1.
J(g(z + 1)) - J(g(z)) = P(z) . We note that whenever g(zo + 1) = g(zo) for some Zo, the function on the left side of the above equation will assume a value zero. Therefore, we must have g(z + 1) - g(z) = PI(z)e adz ) , (3.3) where Pdz) is a polynomial and adz) is an entire function. Similarly, we deduce (3.4) where P2(Z) is a polynomial and a2(z) is an entire function. Substituting z with z + 1 in equation (3.3) and then adding to equation (3.4) we get
g(z + 2) - g(z) = Pdz + l)e adz +l) + Pdz)ea.(z) .
(3.5)
Equating (3.4) and (3.5) we get
Applying Borel's lemma, we conclude that
adz
+ 1) = adz) + c ,
c a constant. We write
Then
Thus H2(Z) is a periodic function with period 1. We easily verify (for instance, by equating the coefficients) that for any given polynomial Pdz) there always exists a polynomial q(z) such that
eCq(z + 1) - q(z) = Pdz) .
112
Fix-points and Factorization of Meromorphic Functions
Setting
Hdz) = g(z) - q(z)e adz ) = g(z) - q(z)e H2 (z)+cz ,
(3.6)
we obtain
Hdz + 1)
g(z + 1) - Pdz + 1)e H2 (Z+I)+CZ+C = g(z) + q(z)e H2 (z)+cz - eCq(z + 1)e H2 (z)+cz = g(z) - (eCq(z + 1) - q(z))e H2 (z)+cz = g(z) - PI (z)e H2 (z)+cz = HI(z) .
=
Hence HI (z) is also a periodic function with period 1. It follows from (3.6) that and the theorem is thus proved.
Remark. (i) H F is periodic mod a nonconstant entire function h(z) with p(h) ~ 1, then the theorem remains valid, where q(z) need not be a polynomial, but p(q) ~ 1. (ii) In general, it seems there is not much that we can really say about the factors I and g in a factorizatio~ F = I(g). However, for functions F of certain forms, we can determine the possible forms of I or g in the factorization F = I(g). Gross, Koont and Yang proved: Given F(z) == HI(Z) + ze H2 (z), where HI and H2 are periodic entire functions with the same period T. For any factorization F(z) = I(g(z)), where I, g are entire functions with I being nonlinear, then g(z) must be of the form g(z) = T(z) + az, where T(z) is a periodic entire function with period T and a is a non-zero constant. Furthermore they also proved if H2 is prime, so is F. Theorem 3.4. H P(z) is any nonlinear polynomial and g(z) is an arbitrary transcendental entire function, then P(g) is not periodic mod a non-constant polynomial. Proof. Assume the theorem to be false; then there exists a non-constant polynomiall(z) such that
P(g(z + 1)) - P(g(z)) By the above theorem, g is of the form
=
l(z) .
(3.7)
Factorization of Meromorphic Functions
113
where Hi(Z),i = 1,2 are periodic entire functions with period 1, q(z) is a polynomial, and c is a constant. Substituting z by z + 1, z + 2, ... ,z + n - 1 successively in (3.7) and adding them up, we obtain
P(g(z + n)) - P(g(z)) = l(z)
+ l(z + 1) + ... + l(z + n -
1) .
(3.8)
Alternatively, from (3.7) we can derive
g(z + n) = g(z)
+ [q(z + n)e cn - q(z)Je H2 {z)+cz
•
Substituting this into (3.8), we get
P(g(z) + [q(z + n)e cn - q(z)]e H2 {z}+cz) = P(g(z)) + l(z) + ... + l(z + n -1). (3.9) IT leci < 1, then at z = 0, the right side of (3.9) tends to infinity with n, whereas the left side is bounded. This is a contradiction. When leci > 1, replacing z by -z in (3.9), we will arrive at a similar contradiction. Hence we may assume that leci = 1. Let t be the degree of l(z). When z = 0 and n is sufficiently large, then the absolute value of the right side of (3.9) is greater than
but less than where .Al,.A2 are suitable constants. Assume the degree of P = u and the degree of q = v then from (3.9) we conclude that uv = t + 1. Now we shall treat the two cases separately: (a) a(z) = eH2 (z)+cz = a, a constant and (b) a(z) is not a constant. IT case (a) holds, then
P(H1 (z)
+ aq(z + 1)) - P(Hl(Z) + aq(z)) = l(z) .
For large r(= Izl) the maximum modulus of the left side of the equation is greater than .A(M(r, Ht})u-l for some positive constant .A. This is impossible, since u 2:: 2 and HI (z) is transcendental. Now assume that case (b) holds. Then it can be easily verified that for every z the right side of (3.9) is less in absolute value than .Ant+l for some positive constant .A (independent of z) and sufficiently large n > N(z)j N(z) is a quantity that depends on z. In order to estimate the left side of
114
Fix-points and Factorization of Meromorphic Functions
(3.9), we may assume without loss of generality that P(z) and q(z) assume respectively the following forms:
P(z) = >'uzu + ... q(z)=zt/+ ... We can choose Zo so that
Then for any 0 < e < 1 and sufficiently large n (depending on N(zo) and e), the left side of (3.9) is greater in absolute value than
(3.11) It follows from this and (3.10) that
>'nt+1 > (1 - e)2>.nt+ 1
,
for sufficiently large n and any small e. This will lead to a contradiction and the theorem is thus proved.
Remark. The theorem remains valid if the order of 9 is assumed to be less than 1, then the function q( z) in the expression of g( z) will satisfy
p(q) ::; 1. Theorem 3.5. If P(z) is a polynomial of degree> 2 and if f is a transcendental entire function, then f(P) is not periodic mod q (q a polynomial). Proof. Suppose that the theorem is not true. We may then assume that the function F(z) == f(P(z)) has a pseudo-period i such that
F(z + i) - F(z) = '1, >'2, >'3 and >'4 such that
>'1 0 Pm 0 >'2(U) = Tm , >'10 Pn 0 >'3 = Tn, >, 2 1 0 Un 0 >'4 = Tn, >.;10 Vm 0 >'4 = T m , >'10 Pm(Un ) 0 >'4 = >'1 0 Pn(Vm ) 0 >'4 = Tnm , where the Tk denotes kth Chebyshev polynomial. (B) Suppose m > n. There exists linear polynomials >'1, >'2, >'3, >'4 and a polynomial h of degree less than such that
min
>'10 Pm 0 >'2(U) = u 2h(u)n (r + ndeg h >'2 1 Un 0 >'4(S) = sn , >'10 Pn0 >'3(U) = un , >.;10 Vm 0 >'4(S) = Sr h(sn) ,
>'10 Pm (Un) 0 >'4(S)
= m)
,
= >'1 0 Pn(Vm) 0 >'4(S) = [Sr h(sn)t
.
Proof. Without loss of generality we may assume, 1 < n1 < q < n2 < n3 < ... ; nj+1
~
qnj, i
=
1,2, ...
(3.28)
Applying Lemma 3.5 with m = njU ~ 2), n = q, we get polynomials U (of degree q), V (of degree m), and entire function sm(z) such that:
(3.29) Now according to Lemma 3.7, the polynomials Pm, U, Pq , and V must satisfy one of the cases, (A) or (B). We shall show that if case (B) holds for a certain pair (m = nj, n = q), then (B) will hold for any other pairs (nk' q). Alternatively suppose that we consider case (A) for a certain pair (nk' q). There exists linear polynomials p and (1 such that, po Pq 0 (1(u)
= Tq(u) ,
where Tq is qth Chebyshev polynomial. On the other hand, case (B) holds for m and n = q < m. Thus there are linear polynomials J.L, >. such that
Hence
Factorization of Meromorphic Functions
129
or
A(Bu + C}q
+D =
Tq(u} .
Note that Tq does not assume any value of multiplicity ~ 3. The above equation leads to a contradiction, since q ~ 3. Therefore, Theorem 3.12, will be a consequence of the results of the following two results, Theorems 3.13 and 3.14.
Theorem S.lS. IT there exists an infinite sequence M = {mk} and integer q ~ 3 satisfying (q, mk) = 1 (for k = 1,2, ... ) such that (3.29) and case (A) of Lemma 3.7 hold, then F must have the form: F(z} = acos VH(z} + b; where a(¥= o} and b are constants, and H(z} is an entire function. Theorem S.14. IT there exists a sequence of values of m( = {mk}} and n = q satisfying the condition of Eq. (3.28) such that Eq. (3.29) and case (B) in Lemma 3.7 hold, then F must have the form:
F(z}
= aeH(z)
+b
where a(¥= o} and b are constants and H(z} is an entire function.
Proof of Theorem S.lS. By case (A), corresponding to each m EM, there exists linear polynomials .Am and 11m such that
IT ~m and £1m are the linear polynomials that correspond to we have \ ,-1 Am 0 Am 0
T.q
--1 Ollm 0 11m
m EM.
Then
= T.q.
Recalling that for q ~ 3,Tq(u} cannot be expressed as A(u - a}q + B, from this and Lemma 3.5, we can conclude that there can only be a finite number of pairs (.Am 0 ~;;.1, £1;;.1 Ollm ). Keeping m fixed and replacing M by an infinite subsequence N, if necessary; we may assume that .Am does not depend on the choice of m. It follows from Eq. (3.29) and (A) that there exists a linear polynomial .A such that
where Sm(Z} is a composition of the function sm(z} in Eq. (3.29) and a linear polynomial.
130
Fix-points and Factorization of Meromorphic Functions
Setting (3.30) we have ). 0
F(z) = cos(qmtP(z)) .
(3.31)
The expression tP in Eq. (3.30) is a multivalued function but in a disk D (in the z-plane), that contains no roots of srn(z) = ±1, we can define tP without ambiguity as a specific branch of cos- 1 srn(z). IT tPo is one of the branches of cos- 1 srn(z), then any other branch tP(z) is determined by the formula: tP(z) = ±tPo(z) (mod2n-), Vz ED. Let n be a member in N other than m. We can obtain functions sn(z) and ,p(z) having properties similar to that of srn(z) and tP(z) respectively such that
). 0
Sn(z) = cos,p(z) ,
(3.32)
F(z) = cos(qn,p(z)) .
(3.33)
Similarly, for the multivalued function ,p, we can define in a disk (that incidentally can be chosen to be identical to D) a specific branch ,po such that ,p(z) = ±,po(z) (mod 21r), Vz ED. (3.34) From Eqs. (3.31) and (3.33) we have
qn,po(z) = ±qmtPo(z) (mod 21r) . We may suppose (if necessary, by changing ,po into -,po)
qn,po(z) Hence,
=
qmtPo(z) + 2k1r .
m 2kn,po(z) = -tPo(z) + .
n
We set ~o(z) -
= ,po(z) + 2tn-, m n
,po(z) = -tPo(z) +
Cj
(3.35)
qn
where t is suitable integer, such that C
is a constant satisfying - n-
'o(t) = 0 and >'k(t) = 0 respectively, we shall show that or = /3. Assume that Q::f /3, then by Nevanlinna's second fundamental theorem, we have
for a suitable sequence of r values tending to 00. On the other hand, we observe that each root of the equations, F(z) = and F(z) = /3 has multiplicity of at least nl and q, respectively. Hence
Q
-( 1) ::;-N 1 (r'-F 1) ::;-(I+o(I))T(r,F) 1 N r'-F , -
and
N (r, F
Q
nl
-
Q
nl
~ /3) ::; ~N (r, F ~ /3) ::; ~(1 + o(I))T(r, F)
.
Substituting the above two results into Eq. (3.39) and noting that N(r, F) = 0, we get
T(r, F) ::;
(~+ -..!.. q nl
+0(1)) T(r, F) ,
which is impossible. It follows that Q = /3, and hence >'k(t) = c>'o(t). Replacing Sk (t) by 6Sk (t), if necessary, we may assume that c = 1, i.e.,
>'0
=
>'k
(k = 2,3, ... ) .
Factorization of Meromorphic Functions
133
Thus
(3.40) By assuming (n1' q) = 1, we easily see from the above expression that the multiplicity of every root of Fo(z) = 0 is divisible by n1q. Hence, we have
where G is an entire function. From this and Eqs. (3.38) and (3.40), we have
We let a
where J.L( a) denotes the multiplicity of the root anI. Then n,
h(t n ,) =
II II (t a
p;a)l'(a)
;=1
where p is a primitive n1-th root of unity. IT Zl is a zero of s(z) - ria with multiplicity /.I, then ql/.lJ.L(a) Suppose that qf'J.L(a) then /.I ~ 2, and ria is a completely ramified value of s(z). Also r + n 1deg h = q,O < r < q. It follows that every root of s = 0 has a multiplicity ~ 2. Thus if deg h > 0 and h has some zero with a multiplicity not divisible by q, then s has at least three completely ramified values. This is impossible. Hence we see that there are only two possible cases that arise
(i) srh(snl) = sq or (ii) qlJ.L(a) for every zero of h .
(3.41)
However, case (i) implies srh(snl) = sr(H(sn,))q where His a polynomial. It also implies that q = r + n1 deg h = r + n1q deg H. This is impossible for deg H > o. Thus only case (ii), i.e., Eq. (3.41) can hold. Therefore, Eq. (3.37) becomes
Fo(z) = s(zt,q . Letting k
= 2 in
Eq. (3.40), we get
134
Fix-points and Factorization of Meromorphic Functions
and we may also assume
Substituting 8 by 82 and G by 8, and exchanging the positions of q, we can, using the reasoning above, obtain
where K2 conclude
IS
a polynomial Since
r2
> 0 and n2
~ nlq, (n2' q)
nl
and
1, we
Hence,
By repeating this kind of argument, we are led to
This shows it is impossible for Fo(z) to possess any zeros_ Thus
Fo(z) = >'oF(z) =
eH(z)j
H(z) an entire function.
Since we have shown that F has form (3.23)' Theorem 3.14 is proven. This also concludes the proof of Theorem 3.12. 3.5. FACTORIZATION OF ELLIPTIC FUNCTIONS In the previous section 3.4 we discussed the possible factors of the periodic cosine function and realized that they are quite restricted. Now we shall study the possible factors of an elliptic function h(z). More specifically, we would like to find when h(z) = J(g(z)) what forms and properties, J (left factor) and 9 (right factor) may possess.
Theorem 3.15. factor.
No elliptic function h(z) may have a periodic left
Proof. First we recall that the order and lower order of an elliptic function h are both equal to 2. This is because any elliptic function can be expressed as a rational function of a sigma function and its derivatives.
135
Factorization of Meromorphic Functions
Now the sigma function (and hence its derivatives) has order 2, so that the order of h (= p(h)) ~ 2. On the other hand, to any value "a",
T(r_1 »N(r_1»~ ,h- a
-
,h- a
log r
for some constant c. This leads to the conclusion that p(h) 2: 2. Therefore we arrive at p(h) = 2. We now suppose that h is not prime and has the factorization h = fog. IT f and 9 are both transcendental entire functions then, by P6lya's theorem, p(j) = 0, and f cannot be a periodic function. Also if f is periodic, say period 1, then 9 must be a polynomial. Consider a point set S = {zlz is a root of one of the equations g(z) = m + c, m = 0, ±1, ±2, ... j c is a constant}. We shall show that S has one finite limit point. Let g(z) = AkZk+ ... +A1Z+Ao and Zm,Zm+j be the roots satisfying g(z) = m + c, g(z) = m + i + c respectively. We have
Ig(zm) - g(zm+j)1 = IZm+j - zmIIAkllz~-~/j + ZmZ~+2j + Z~-l + Pk- 2(Zm, zm+j) I = i ,
+ ... (3.42)
where Pk - 2 is a polynomial with variables Zm and zm+j of degree k - 2 at most. We can easily see that for each m, there always exists integers il, J2 2: m such that lil - J21 < 4k2 and I arg Zl - arg z21 ~ ~. We can thus derive
which approaches 00 as m --+ 00. This would imply from Eq. (3.42) that IZm+j - Zm I tends to as m --+ 00. Therefore, S must have a limit point. Now we have h = foP where P is a polynomial. Assume that h has periods Tl and T2 and recall that f has period 1, then the following identity holds for any integers m, nl and n2.
°
(3.43) For any fixed Zo we set
From the above analysis, we see that S has a finite limit point. Thus Eq. (3.43) will yield a conclusion that f is a constant, which is a contradiction. The theorem is thus proven.
Fix-points and Factorization of Meromorphic Functions
136
With regard to the right factors of elliptic functions we have the following result.
Theorem 3.16. Let h(z) be an elliptic function of valence 2. If h = fog for some transcendental entire function f and an entire function 9 that is not a linear polynomial, then the right factor 9 must be either (i) a polynomial of degree 2 or (ii) of the form A cos(z + r) + B, where A, B, and r are constants. Proof. By assuming that h is an elliptic function of valence 2, it means that h satisfies the following differential equation:
(h')2
=
P(h) ,
where P is a polynomial. Thus we have
(g' /,(g))2 = P(I(g)) . If follows that '2
9
=
P(I(g)) (I' (g))2
=
F( ) g,
where F(~) = P(I(d)/ f'(d 2, a meromorphic function. According to a theorem of Clunie's, we conclude that F(~) must be a ra. If . 0 h . 1· T(r,F(g)) ·11 1· -1· T(r,F(g)) tiona unction. t erwlse 1m T( ) = 00 WI resu t In 1m T( 2)
=
r-+oo
r,g
r-+oo
r,g'
00.
Therefore, we have c is a constant
=1=
0 .
(3.44)
Note that since g' has no poles, the denominator in the above expression never vanishes. Moreover, by examining the multiplicities of the roots of g(z) = ai, we can verify easily that 9 has at most two complete ramified values, say al and a2. Then Eq. (3.44) becomes
From this we conclude that if 9 is a polynomial then deg 9 = 2, and nl = = 1 if 9 is a transcendental entire function. In the latter we have
n2
137
Factorization of Meromorphic Functions
gl2 = ddg(z) + d2)2 + d3 , where d1 ,d2, and d3 are constants. It follows that g(z) has the form Acos(cz + r) + B. This also completes the proof. Remark. It is not difficult to exhibit some elliptic functions have transcendental right factors. Sn(z), Cn(z) and dn(z) are such functions. •
Sn(2kz/7r) = csmz
II 00
(
n=l
1 _ 2q2n cos 2z + q4n ) 2 1 4 2 1 - 2q n- cos 2z + q n-
We easily see that Sn(2kz/7r) = J(sinz)' where
J(d = c~
II 00
(
1- 2q2n(1- 2~2) 1- 2q2n-l(l- 2~2)
+ q4n ) + q4n-2 '
a meromorphic function.
Earlier in this chapter, we showed that if J is a transcendental entire function and P an arbitrary polynomial of degree ~ 3, then J 0 P cannot be periodic. We conclude this section by proving the following result.
Theorem 3.17. Let J(z) be a non-constant meromorphic function and P(z) be a polynomial of degree n. Then F(z) = J oP(z) cannot be periodic unless n = 1,2,3,4 and 6. Proof. It is, of course, possible for n = 1 and 2. Therefore we shall only deal with the cases when n ~ 3. Suppose that F is a periodic function and by changing variables if necessary, we may assume without loss of generality that F has a period of 1. Moreover, we may assume P(z) has the form
P( Z ) = aoz n + an-tZ n-t + an-t-lZ n - t - l + ... + ... where t is an integer equation
~
2. It is clear that for any given z and the following
P(~)
= P(z + m)j
JzJ > ro ,
(3.45)
always has a root. Furthermore, for sufficiently large m, we have ~
= ,,(z + m) + 0(1) (m -+ 00) ,
(3.46)
where" = e21ri / n • We observe that, for sufficiently large m, any integer m ' (from the above we have k + m'l) will be greater than ro as in Eq. (3.45). Since F has period 1, F(~)
=
J(P(~)) =
J(P(z
+ m)) = f(P(z))
=
F(z) .
Fix-points and Factorization of Meromorphic Functions
138
On the other hand, to the ~ in Eq. (3.46), the equation P(~') = P(~
has a root ~' satisfying ~' m -+ 00. Thus F(~'
=
11(~
+ m')
+ m') + 0(1) = I1 2 z + 11 2 m + 11m' + 0(1)
as
+ m) = F(~') = F(d = F(z) .
Consequently to a given point z, the equation
F(w) = F(z)
(3.47)
always has a root w satisfying
for any given integer m' and Iml sufficiently large. Since 11 = e21ri / n , we have
Suppose that 2cos
2: is a irrational number.
To any given real number
f3, we can choose m and m' such that the right-hand side of the above expression can be made arbitrarily close to f3. Thus F(I1 2 z
+ 11f3) = F(z) (-00 < f3 < 00) .
However, z is an arbitrarily given number, hence, from the above equation, F must be a constant function. This creates a contradiction. Suppose that cos = a is a rational number. Then the nth root of unity 11 will satisfy the equation
2:
In the meantime, 11 must satisfy an irredicuble equation g(I1) = 0, where the degree of g is cp( n)( cp( n) denotes the Euler function of n). Therefore, 112 - 2al1 + 1 must be divisible by g(I1). Hence cp(n) = 1 or 2_ However,
cp( n) = n
II (1 qln
~) ~ II (q gin
1) .
Factorization of Meromorphic Functions
139
It follows that if ip(n) ~ 2, then n can only have 2 and 3 as its factors. This results in n = 3,4 or 6. The theorem is thus proved.
Discussion. Illustrate by examples, that for n = 3,4 or 6 there exists a meromorphic function In and polynomial Pn(z) of degree n such that
In(Pn(z)) is an elliptic function. 3.6. FUNCTIONAL EQUATIONS OF CERTAIN MEROMOPRHIC FUNCTIONS Factorization theory can be included in the theory of functional equations. The factorization of F(z) = I(g(z)) can be viewed as the finding of functions F, I and 9 that will satisfy the expression just mentioned. Various forms of functional equations have been derived in the course of studying problems relating physics, practical or theoretical mathematics. For example, a well-known problem is Cauchy's functional equation: I(x + y) = I(x) + I(Y). In general, it is difficult to obtain a concrete solution to a functional equation. Many have obtained results and focused their research on the necessary and sufficient conditions for the existence of solutions or certain special properties of the solutions. Here we shall introduce certain simple forms of the functional equations of meromorphic functions to show the existence as well as the growth properties of the solutions. We shall first discuss the following type of equation:
I(g(z)) = h(z)/(z) , where we restrict following results:
I, 9
(3.48)
and h to entire functions. It is easy to derive the
Theorem 3.18. Let I(z), g(z) and h(z) be entire functions and satisfy the Eq. (3.48). Suppose that h(z) is a polynomial and 9 is not a linear polynomial. Then I must be a polynomial. Theorem 3.19. Let I(z), g(z) and h(z) be entire functions and satisfy Eq. (3.48). Suppose that both I and h are non-constant polynomials, then g must be be a polynomial. Theorem 3.20. Let g(z) == z2 and I, 9 be non-constant entire functions. Assume that h has only a finite number of zeros. If Eq. (3.48) holds for such I, g, and h, then I must have only a finite number of zeros as well. Most of the results introduced here were obtained by R. Goldstein who also extended the previous discussion by considering meromorphic solutions
140
Fix-points and Factorization of Meromorphic Functions
of the following type of equation:
I(g(z)) = h(z)/(z) + H(z) _
(3.49)
Theorem 3.21. Let I, g, hand H be meromorphic functions satisfying Eq. (3.49). Suppose that I, g are non-constant functions and g is always a transcendental entire function unless I is a rational function_ Also suppose that there exists a positive constant k such that for r> ro (a constant),
T(r, h) T(r, H)
~ ~
kT(r, f) , kT(r, f) .
(3.50)
Then g must be a rational function of, say, order m_ Furthermore, if transcendental, then 1 ~ m ~ k + 1, and when m > 1, I must satisfy
where e is any given positive number and a
I
is
= log(2k + l)/logm.
Proof. From Eqs. (3.49) and (3.50), we have
T(r,J(g))
T(r,f) + T(r,h) + T(r, H) + 0(1) < (2k + l)T(r, f) + 0(1) . ~
But Clunie proved that if
I
(3.51)
and g are transcendental, then lim T(r,l(g)) =
r-+oo
T(r, f)
00
(3.52)
which will contradict with Eq. (3.51)_ Hence I and g cannot both be transcendental. It is clear that if I is rational function and g is transcendental, then from Eq_ (3_50) we conclude both hand H are rational functions. As a result Eq_ (3.49) will not hold. For this equation to hold, I must be transcendental and g must be a polynomial. Before proceeding further we prove the following result.
Lemma 3.S. Let !/I(r) be a positive and continuous function of r satisfying, for some m > 1, (3_53)
141
Factorization of Meromorphic Functions
where p., A(A > 1) are two positive constants. Then
¢(r) = O((logr)a)j
Q
= logA/logm.
Proof. We put in Eq. (3.53) log p. 1-m'
¢(r) = 4>(t)
A4>(t),
(t
t = logr - - that yields
4>(mt) We choose becomes
Q
~
~
to) .
such that m a = A and put ¢(t) = 4>(t)/ta. Then Eq. (3.53)
cp(mt)
~
cp(t),
(t
~
to) .
Now, cp(t) is also a positive and continuous function for t > 0 and the above inequality ensures that it is bounded above by some number B for sufficiently large values of t. Thus
¢(r) = 4>(t)
~
Bt a = B (IOgr _ IOgP.)a
~
Bdlogr)aj
1-m
where Bl is a suitable constant.
Now we continue the proof of Theorem 3.21. Let ( ) =amz m +am-lz m-l + ... +ao gz
(am
i- 0),
m ~ 2,
and p. = laml- 6 (0 < 6 < laml). Since Ig(z)1 ,..., lamlrm for sufficiently large values r, we have for any value "a".
n(r, a, f(g))
~
mn(p.rm, a, f),
(r
~
ro) .
By integrating, we get
N(r, a, f(g)) - N(ro, a, f(g))
~
j
r
mn(p.tm a f) " dt + 0(1) log r . r
ro
We put s = p.tm and obtain
j
r
ro
mn(p.tm, a, f) dt = (Wm n(s, a, f) ds t
JI'r'(J'
S
= N(p.rm, a, f) - N(p.rO' , a, f)
+ O(log r) .
Fix-points and Factorization of Meromorphic Functions
142
By combining the above two inequalities, we have
N(J.l.r m , a, I) +O(logr)
~
N(r,a,/(g)),
(r ~ no).
(3.54)
But, it is well known that for a suitable value a,
N(r, a, I) ,... T(r, I) and log r = o(T(r, (3.51) we obtain
I))
for a transcendental function
T(J.l.r m , I)
~
(2k + 1 + e)T(r, I),
I.
By Eqs. (3.54) and
r ~ rl .
Applying Lemma 3.8, the required result follows. By a similar argument we can obtain the following result:
Theorem 3.22. Let I, 9 and h be non-constant meromorphic functions satisfying Eq. (3.48). Suppose that 9 is always a transcendental function unless I is rational. If there exists a positive constant k such that
T(r, h)
~
kT(r, I),
(r
~
ro)
then 9 must be a rational function of order m. Furthermore, if m > 1 and e is any given positive number, then unless I is rational, m ~ k + 1, and
T(r, f) where
f3
=
= O(log r)H..
as r
-+ 00 ,
log(k+ l)/logm, and
T(r,h) --- > m - 1 - e T(r, I)
(r
~
rt} .
(3.55)
In the following we shall investigate Eq. (3.48), in which the zeros or poles of h have been restricted. We shall call the value a a Fatou exceptional value of 9 of multiplicity m if g(z) == a + (z - a)meG(z); where G(z) is an entire function.
Theorem 3.23. Let I and h be meromorphic functions, and 9 be a nonlinear entire function satisfying Eq. (3.48). Suppose that h has no poles (zeros)' then I has at most one pole (zero) at z = 0:, and a is a Fatou exceptional value of 9 of multiplicity 1. Proof. We give the proof for the case where h(z) has no poles. The case where h has no zeros can be proved similarly. Using the assumption,
Factorization of Meromorphic Functions
143
we easily see that if z = a is a pole of J(z) and g(b) = a, then z = b must be a pole of J(z). Repeating this argument yields if z = a is a pole of J and if for some n, gn(z) (the nth iterate of g) = a, then z = b is also a pole of J(z). We now need a result of Fatou's as follows.
Lemma 3.9. Let g(z) be a nonlinear entire function. Then there exists a nonempty perfect set T(= T(g)) of complex numbers with the property that to any Zo E T and an arbitrary number w (with one possible exceptional value) there corresponds a sequence of positive integers {nj} U = 1,2, ... ) and a sequence of complex numbers {Zj} U = 1, 2, ... ) such that limzj
= Zo
and
gnj (Zj) = w,
(j = 1,2 ... ) .
(3.56)
This result combined with the conclusion at the beginning of the proof, implies that J has at most one pole, at z = a. Furthermore, z = a must be a Fatou exceptional value of g(z). Hence we have (3.57) where G(z) is an entire function and m is a non-negative integer to be determined. We express J(z) as
J(z) = F(z - a) , (z - a)n
(3.58)
where F is an entire function with F(O) =I=- 0 and n is a positive integer. We will only treat the case where m ~ 1 (a similar argument applies if m = 0). Then Eqs. (3.48), (3.57), and (3.58) yields
F{(z - a)meG(z)} _ h(z) F(z - a) (z - a)nmenG(z) (z - a)n Consideration of the order of the pole at z = a leads to either n = 0 or m = 1. But when n = 0, J becomes an entire function which contradicts
144
Fix-points and Factonzation of Meromorphic Functions
the hypothesis that z = a is a pole of I. Therefore m = 1 and I and g are given by Eqs. (3.58), (3.57) respectively. Theorem 3.23 is thus proven.
Corollary 3.1. Let I, g and h be as in Theorem 3.23. If g(z) and h(z) are nonlinear polynomials, then I(z) has no poles (zeros). Theorem 3.24. Let g(z) be a polynomial of degree m ~ 2, and let I, h be meromorphic functions satisfying the equation I(g(z)) = h(z)/(z). Suppose that I is of finite order and 6(0, h) = 6(00, h) = 1. Then 6(0, f) = 6(00, f) = 1 and Ph = mPI; where Ph and PI are the orders of I, h respectively and must be positive integers.
Proof. It is well known under the hypotheses that P/(g)
= mPI
.
(3.59)
By 6(0, h) = 6(00, h) = 1, h is of regular growth and is of positive integer or infinite order. Now
h( ) = I(g(z)) z I(z) , hence,
Thus
> 0 and (3.60) Ph ~ mPI < 00 . On the other hand, from the equation I(g) = hi and Eq. (3.59) we have PI
(3.61) Since PI < 00, we deduce from the above inequality that
(3.62) Thus, by Eqs. (3.60) and (3.62), Ph
=
mp I
=
PI(g)
< 00 .
From this and the fact that h is of regular growth we have
(3.63)
145
Factorization of Meromorphic Functions
whence lim T(r, h) T(r, I)
= 00 .
(3.64)
"-+00
From the equation I(g)
= hI,
we have
T(r, log) :5 T(r, h) + T(r, I) . Hence from Eq. (3.64) we get
T(r, log) < 1 + 0(1) T(r, h)
-
(as r _ 00) .
Conversely, from h = I(g)/ I and Eq. (3.64) we deduce
T(r,/(g)) > 1 + 0(1) . T(r,h) Thus, we have
T(r, I(g)) '" T(r, h) . Now n(r, 0, I(g)) :5 n(r, 0, h)
(3.65)
+ n(r, 0, I), so
N(r, 0, I(g)) :5 N(r, 0, h) + N(r, 0, I) + O(log r) :5 N(r, 0, h) + T(r, I) + O(log r) . Using Eq. (3.64) and the assumption 8(0, h)
=
1, we have
-1· N(r,O,/(g)) < -1· N(r,O,h) + (1) = (1) 1m T(r,h) - ,.~~ T(r,h) 0 o.
(3.66)
Therefore, by Eqs. (3.66) and (3.65) we have lim N(r,O,/(g)) "-+00
T(r,/(g))
=
lim N(r,O,/(g)) "-+00
= 0(1)
.
T(r, h)
This shows that 8(0, I(g)) = 1. Similarly we can prove that 8(00, I(g)) = 1. Now we prove that PI must be a positive integer. If log is a meromorphic function of finite order, and g is a nonlinear polynomial satisfying I(g) = hI, then by 8(0, I(g)) = 8(00, I(g)) = 1, we also have 8(0, I) = 8(00, I) = 1. (For the proof, we refer the reader to Goldstein's paper "Some results on factorization of meromorphic functions", J. London Math. Soc. (2).( (1971) 357-364.
146
Fix-points and Factorization of Meromorphic Functions
Hence, PI must be a positive integer. This also completes the proof of the theorem.
Discussion. (i) H g(z) is allowed to be meromorphic and h has no poles, what conclusions will result? (ii) Does the condition 6(0, f(g)) = 6(00, f(g)) = 1 always lead to 6(0, f) = 6(00, j) = 1? Theorem 3.25. Let f be a non-constant meromorphic function, g be an entire function, and q(z) be a polynomial of degree k (;::: 1) satisfying the following equation
f(g) = q(l) . Then q(z) must be a polynomial of degree m ~ k. Furthermore, if m > 1, then T(r, j) = O(l)(log r)a, where ex
=
(log k/ log m)
+ e (e is any given positive number) _
We omit the proof, this being analogous to the proof of Theorem 3.21. 3_7_ UNIQUENESS OF FACTORIZATION For simplicity we shall only discuss entire functions and their entire factors_ We state that two factorizations (of entire factors) fdh(··· (In)) ... ) and gdg2(" . (gn)) ... ) are equivalent if there exists linear transformations .Al, _. - ,.An-l such that
An entire function F is called uniquely factorizable, if all its factorizations of nonlinear prime entire factors are equivalent to each other_ Ritt obtained a complete answer to the uniqueness factorization problem for polynomials (see the appendix). The result essentially states that, besides the following three non-equivalent cases, for pairs of consecutive factors fdh) and gl (g2), the two factorizations of a polynomial F(z) will be equivalent. The exceptions are: (i) h(z) = zk, h(z) = zl and gl (z) = zl, g2(Z) = zk (ii) h(z) = zk[h(z)]l, h(z) = zl, and gdz) = zl, g2(Z) = zkh(zl), and (iii) h(z) = Pk(Z), h(k) = Pl(z), and gdz) = PI(Z)' g2(Z) = Pk(Z), where Pk(Z) is the kth polynomial satisfying cos kz = Pk(cos z)_ Case (ii) may also arise in the factorization of a transcendental entire function. H, for example, we let F(z) be zP exp zP (p is a prime number)'
Factorization of Meromorphic Functions
147
then F has two factorizations that are not equivalent: F(z) = zP 0 (ze ZP Ip) and F(z) = (ze Z) 0 zp. However, it is not difficult to show that both F(z) = zPePZ(= zP 0 ze Z) and F(z) = (ze Z) 0 (ze Z) are uniquely factorizable. Moreover, the latter is almost the simplest function one can show in demonstrating the uniqueness factorization of transcendental entire factors. As a generalization, H. Urabe obtained the following result in his dissertation.
Theorem 3.26. Let F(z) = (ze Z) 0 (h(z)e Z), where h(z) is a nonconstant entire function of order less than one, and has at least one simple zero. Then F is uniquely factorizable. Proof. (sketch) Let F(z) = (ze Z) 0 (h(z)e Z) = I(g(z)); I, g being two nonlinear entire functions. By virtue of the assumption that h(z) has at least one simple zero and the Tumura-Clunie Theorem we conclude that I cannot be a polynomial. According to a result of Edrei-Fuchs' that if I and g are two transcendental entire functions with the exponent of convergence of the zeros of I being positive, then the zeros of I(g) have an exponent of convergence equal to infinity. Therefore, we need only consider three cases: (a) I(z) = hdz)eP(z), hI nonlinear entire function with p(h) = 0, p(z) is a non-constant polynomial, and g(z) is a transcendental entire function with p(g) < 1; (b) I(z) = zeP(z), g(z) = h(z)eq(z), where p, and q are non-constant entire functions; and (c) I(z) = hdz)eP(z), where g(z) is a polynomial of degree ~ 2, and hI and p are non-constant entire functions satisfying p(hd < de~g (hence p(hd(g) < 1). In case (a), from F = I(g), we obtain hl(g(z)) = h(z)ed(z) and p(g(z)) = z - d(z) + h(z)e Z, where d(z) is an entire function with p(d) < 1 (Polya's Theorem, Corollary A.1, Appendix). By Theorem 4.2, it follows that p(z) must be a polynomial. Thus p(g) = 1, which is a contradiction. In case (b), we obtain a functional equation q(z) +p(h(z)eq(z)) = z+h(z)e z . It is easily verified from this relationship that q(z) must be linear and the uniqueness of factorization follows. In case (c), we have the relations hdg(z)) = h(z) and p(g(z)) = z + h(z)e z . Using these equations and an argument similar to the proof of case (b) we easily arrive at the uniqueness of factorization of F. Urabe also obtained the following more general result. Theorem 3.27. Let F(z) = (z + h(e Z)) 0 (z + q(e Z)), where h(z) is a non-constant entire function with the order p(h(e Z)) < 00 and q(z) is a
148
Fix-points and Factorization of Meromorphic Functions
non-constant polynomial. Then F(z) is uniquely factorizable. We note that eZ and cos z are pseudo-prime and have an infinite number of different factorizations. There exist some transcendental entire functions that are not pseudoprime but have an infinite number of equivalent factorizations. For example, if we let F(z) = z - sin z + sin (sin z - z). Then F(z) = 101 = gog where I(z) = z - sin z and g(z) = sin z - z + 2k1r(k integer =I- 0). The following questions are therefore interesting.
Question 1. Do there exist two nonequivalent factorizations 11 0 12 gl 0 g2, where 11, 12, gl, g2 are prime transcendental entire functions?
=
Question 2. (Gross) Do there exist prime nonlinear entire functions h, 12,··· ,1m and gl, ... ,gn with n =I- m such that
h 012
0 ••• 0
1m == gl
0
g2
0 ••• 0
gn?
4 FIX-POINTS AND THEORY OF FACTORIZATION
4.1. THE RELATIONSHIP BETWEEN THE FIX-POINTS AND THEORY OF FACTORIZATION We have shown in the previous chapter that eZ + z is prime. Gross conjectured that functions F(z) of the form
F(z) = Q(z)e(z)
+z
,
(4.1)
where Q(z) is a polynomial and a(z) is a non-constant entire function, must be prime. To date, the conjecture has not been answered. * However, some partial results have been obtained. Most of these were stated in terms of fixpoints. Recall that at the beginning of Chapter 3 we proved that if P(z) is a nonlinear polynomial and 1 is a transcendental entire function, then P(I(z)) has an infinite number of fix-points (Theorem 3.2). We now prove the following lemma.
Lemma 4.1. Let
1
and 9 be two non-constant entire functions. If
I(g) has only a finite number of fix-points then g(l) also has only a finite number of fix-points.
Proof. Let Zo be a fix-point of I(g). That is, if I(g(zo)) = Zo, then g(l(g(zo)) = g(zo). Thus g(zo) is a fix-point of g(l). Moreover, if Zl and Z2 *Recently (1988) W. Bergweiler confinued this (and hence conjecture 1 in next page) in his preprint entitled "Proof of a conjecture of Gross concerning fix-points" by utilizing Wiman- Valiron type of argument.
149
Fix-points and Factorization of Meromorphic Functions
150
are two distinct fix-points of I(g), then g(Zl) and g(Z2) will be two distinct fix-points of g(f)- IT g(zd = g(Z2), then ZI = f(g(zd) = l(g(Z2)) = Z2This proves the lemma_ From this lemma and Theorem 3_2 we conclude that I(P(z)) has an infinite number of fix-points for any transcendental entire function I and nonlinear polynomial P(z)_ We also know that if I, g are two transcendental entire functions, then either g or I(g) must have infinite number of fix-points. It is easy to see then that Gross' conjecture is equivalent to the following conjecture:
Conjecture 1. IT I and g are two nonlinear entire functions, with at least one of them being transcendental, then I(g) must have an infinite number of fix-points. 4.2. CONJECTURE 1 WITH p(f(g))
1(Z) _ a"
r --+
00 .
O(log r)
(4.21)
158
Fix-points and Factorization of Meromorphic Functions
Next we prove
T(Art, log) = O(T(r, j)),
r -+
00 •
Letting we have
w= z
(1 + ~ + ... + Ck k) t . CoZ CoZ
Since the radical tends to 1 as z -+ 00, it follows that in the region around 00, a single-valued branch of the radical function can be selected. Therefore w(z) becomes an analytic function in the domain {JzJ ~ ro}. Let Art> 2r. The image of the circle {JzJ = Ar t }, under the mapping w (that is 1 - 1 now), will be some curve "Ir lying in the ring:
for some positive constant d. Set
and
h(z) = I(g(z)) = I(coz k +
ClZ k - l
+ ... + Ck) .
Since a point Wo E "Ir corresponds to each point Zo E {JzJ = Art} so that h(zo) = 11 (wo), we deduce
T(Art,h):::; 10gM (Art,h) = log
Mhr, Id :::; log M(Art + d, Id ,
where Mhr,II) = maxJII(z)J. Using the well-known inequality between zE"lr
T(r, j) and log M(r, I), we have T(Art,/(g)) :::; 3T(2(Art + d), 11) . From T(r,/(cz k )) = T(JcJr k , j) and the fact that the Nevanlinna characteristic function T is an increasing function of r, it follows that
T(Art, I(g)) :::; 3T(Jc oJr k (Ar t + d)k, I) :::; 3T(Bl r,j), r -+ 00, where Bl is a suitable positive constant.
(4.22)
Fix-points and Theory of Factorization
159
On the other hand, we have
T(2Arf, J(g)) = T(2Arf, J2) ~
1
1
1.
3" log M(Ari", h) = 3" log M(-y,., il)
~ ~ log M(Arf 1
- d, Jd k
1.
= 3"T(lcol(Ark - d) ,j)
~ ~T(Arf ~
- d, Jd
1
3"T(B2r, J) , (4.23)
where B2 is a positive constant. By Eq. (4.14) and letting deg P
T(r, J(g))
= t,
we have
= T(r, F) = (1 + o(1))B3rt, r -+ 00
,
where B3 is a positive constant. Therefore,
It follows from Eqs. (4.22) and (4.24) that
T(r, j) = O(rt) as r -+
00 •
It follows from Eqs. (4.22) and (4.23) that
T(Ar f , J(g)) = O(rt) = O(T(r, j)), r -+
00 •
Combining this result and Eq. (4.21) yields
T(r,4>l(U)) =o(l)T(r,j) ,
r-+oo.
(4.25)
Let OJ be a zero of the function J(g(U(z))) -4>l(Z), and let IOjl = rj. Then = U(Oj) will be a zero of J(g(z)) - 4>t{z). It can easily be verified, from Eqs. (4.19) and (4.20), that Itjl ::; Art. Therefore
tj
n (Art, J(g) 1_ 4>J
~ n (r, J(g(U)) 1_ 4>l(U)) = n (r, J(z) _l4>d U ))
160
Fix-poin ts and Factorization of Meromorphic Functions
and hence
Thus, by an application of Lemma 4.5, we have (4.26) On the other hand, combining Eqs. (4.13)' (4.14)' and (4.24) yields
N (Art, f(g)l_
~ T(Arf, ~2)
~J
-
N (Art,
:J
+ 0(1) = O(l)T(Arf, f(g))
= o(l)T(r, J) . Comparing this result with Eq. (4.26)' we conclude that s ~l(U(Z)) is, in fact, an (single-valued) entire function. Set ~dU(z)) =
0) and mo = min{mI,m2, ... ,mk,minzER If(z)I}, we have mo > o. Therefore, If(z)1 < mo for some z E K, implies that the point z must li( outside the circles Gi , i = 1,2, ... ,k. When t ~ to then g(h(t)) E K and If(g(h(e))) I < m. This means that when t ~ to, g(h(t)) lies inside some circle Gi . The set {g(h(t)) It ~ to} is connected. It follows from this and the fact that the sets Gi(i = 1,2, ... ,k) are mutually separated that there exists some positive integer, j(1 $ j $ k) such that the following inequality holds:
Ig(h(t)) -
O:il
0 and z E L with Izl ~ ro,
Ig( z) - 0: I $
Vz E ,i, j ~ jo .
e,
(4.47)
Assume that the multiplicity of the zero point 0: is s(~ 1). It follows that there exists a positive constant A (> 0) such that whenever Iz - 0:1 $ e,
If(z)1 ~ Alz that is, whenever Ig(z) -
0:1
0:1" ,
(4.48)
$ e,
If(g(z))1
~
Thus, for z E 'i,j ~ jo (or z E L,
IF(z)1
~
Izl
Alg(z) -
0:1" .
(4.49)
~ ro)
Alg(z) -
0:1" .
(4.50)
170
Fix-points and Factorization of Meromorphic Functions
Since the inequality in Eq. (4.45) holds for z E "ti' we obtain 8
log Ig(z) -
0:1
+ log A
11"
~ - 16 T(lzl, F) .
Consequently, for z E 'Yi(j ~ 30), we have log +
I z 1- I g
()
~ log
0:
I
g
()1 Z
-
0:
I
11" -T(lzl, F) + -log8 A . 168
~
When i ~ 3"a with z = re iO from the above results and by applying Nevanlinna first fundamental theorem, we obtain, for an integer p > 0
T(ri' g)
+ 0(1)
~ m(ri' 0:, g)
> -1
!
- 211"""i ~
Since T(r, F)
-+ 00
as r
1 1211" log+ I (-0) 1 = -211" 0 g reI
log+
-+ 00,
0:
IdB
Ig(re'O)_1 - IdB 0:
11" -1 ( -T(r-,F) 3p
-
168
J
A) .
log+8
(4.51)
the above equation yields
T(ri' g) ~ BT(ri' F),
i
~
30 ,
( 4.52)
where B is a suitable positive constant. However, according to a theorem of Clunie's, for any two transcendental entire functions g and f, lim T(r, f(g)) = r-+oo
T(r, g)
00 .
(4.53)
This contradicts with Eq. (4.52). We must conclude that it is impossible for both f and g to be transcendental entire in the factorization F = f(g). This completes the proof. Remark. Goldstein remarked that Theorem 4.5 remains valid under either of the following two conditions: (i) 6(0, F') = 1 (ii) Ea;o!oo6(a,F) = 1. It was also remarked that the Edrei-Fuchs' result applies not only for 6(a, F) = 1 but also for 6(a, F) > 1- e(p), where e(p) is a positive constant
171
Fix -points and Theory of Factorization
(0 < e(p) < 1) depending on the order of F_ The above remarks also lead to an interesting conjecture as follows_
Conjecture 2. (Fuchs) Let F be an entire function of finite order. IT 8(a, F) > 0 for some complex number, then F is pseudo-prime. Using an argument similar to that used for the preceding theorem the following result can be obtained.
Theorem 4.6. (Gross and Yang) Let P(z) be a polynomial of degree t (~ 1) and hl(Z) and h2(Z)(t 0) be two entire functions of order less than t. Then F(z) == h 1 (z)e P(z) + h2(Z) is pseudo-prime. Hint: Write F(z) as h2(z){~!!=leP(z) + 1}. Question. Does the theorem remain valid if only T(r, hd r -+ oo,i = 1,2 is assumed?
= o(1)T(r, eP )
Recall that a transcendental entire function F is called left-prime or E-Ieft-prime if F = f(g) with f and g being entire implies that f must be linear whenever g is transcendental. F is called right-prime or E-rightprime if F = f(g) with f and g being entire implies g must be linear whenever f is transcendental. Clearly we have (i) IT E is both right and left-prime, then F is E-prime. (ii) A left or right-prime transcendental entire function must be a pseudo-prime. We now provide some criteria for left-primeness.
Theorem 4.1. (Ozawa) Let F(z) be an entire function of finite order whose derivative F' (z) has an infinite number of zero. Suppose for any complex number c, the following simultaneous equations:
{ F(Z)=C F'(z) = 0
(4.54)
have only a finite number of solutions. Then F(z) is left-prime.
Proof. Suppose that F has the factorization F = f(g)j with f and g being transcendental entire functions. From Polya's theorem we must have p(F) = p(F') = O. Hence f'(d has an infinite number of zero, that can be summarized as {~i }~1' There must be some fixed ~i such that the solution to the equation g(z) = ~i are an infinite set. Let {Zn}:=l be the set. The
172
Fix-points and Factorization of Merom orphic Functions
simultaneous equations
{ F(zn) = J(g(zn)) = J(~j) = c F'(zn) = J'(g(zn))g'(zn) = 0,
n
=
1,2, ...
have an infinite number of solutions. This is a contradiction to the hypothesis. We conclude that F must be pseudo-prime. Assume that F = P(g)' where P is a nonlinear polynomial and g is an entire function. P'(d has at least one zero, ~. IT g(z) = Q results in an infinite number of solutions, then using the same argument as above we will get a contradiction. Now assume that g(z) = Q only has a finite number of solutions, this results in
g(Z) =
Q
+ Q(z)eq(z)
,
where q and Q are polynomials. This gives g'(z) a finite number of zeros. Since the assumption states that F'(z) = P'(g(z))g'(z) has an infinite number of zeros, it follows that there must exist a root of P'(~), {3, not equal to Q, such that g(z) = {3 has an infinite number of solutions. This will again lead to a contradiction. The theorem is thus proved.
Exercise. Prove that F(z) = eZ + P(z), where P is a polynomial, is left-prime. Use this result to show that F is E-prime. Exercise. Illustrate the requirement that F'(z) has an infinite number of zeros is a necessary condition for the validity of Theorem 4.7. When no restriction is imposed on the order of F(z), the following results.
Theorem 4.8. (Ozawa) Let F(z) be a transcendental entire function with N (r, },) ~ kT(r, F) for some positive constant k. IT for any complex number c, the system of Eqs. (4.54) has only a finite number of solutions, then F is left-prime. Proof. Suppose that F = J(g), where J and g are both transcendental entire. Finally we assume that J' (~) = 0 has no roots at all. Then
N (r, ;,) = N (r, :,) ::; T(r, g') + 0(1) ::; (1 + e)T(r, g), n.e.,
(4.55)
where "n.e." means the inequality holds nearly everywhere for sufficiently large values of r except possibly a set of r values of finite length.
173
Fix ·points and Theory of Factorization
On the other hand, for any positive integer p and some constant A (not a Picard exceptional value of J), we have
T(r, F)
~ N (r, F ~ A) + 0(1) ~
t
N (r,
3=1
~
(p - 1)T(r, g)
g! 0') + 0(1) , 3
+ O(log rT(r, g))
n.e. ,
(4.56)
where OJ E 1-1 (A). The combination of Eqs. (4.55) and (4.56) yields a result that will contradict the hypothesis of the theorem: N (r, ;,) ~ kT(r, F). If we assume that I has only one zero, ~o, and g( z) = ~o has a finite number of roots, it follows that
N (r, ;,)
= N (r, ;,) = O(log r) :::; T(r, g') + O(log r) :::; T(r, g) + O(log rT(r, g))
n.e ..
This leads to the same contradiction found in the previous case. Alternatively we assume that I'(d has only one zero, ~o, but g(z) = ~o has an infinite number of roots, {Zj}. Then the following simultaneous equations
{ F(z) = I(~o) F'(z) = 0 have an infinite number of solutions {Zj}. This is also a contradiction to the hypothesis. Now we assume that f'(d has at least two distinct zeros. By choosing one of the roots, ~1 so that g(z) = ~1 has an infinite number of roots, we will arrive at the same contradiction. This also proves that F is a E-pseudoprime. Finally, we assume that F = P(g), where P is a polynomial and g is a transcendental entire function. If P' (~) has only one zero and g(~) = ~o has a finite number of roots, then
g(z)
= ~o
+ Q(z)eG(z}, g'(z)
=
(Q'
+ G'Q)eG(z)
where Q is a polynomial and G is an entire function. Then
N (r,
-i) : :; N (r,
Q' +1 G'Q) =
oT(r, g)
n.e ..
(4.57)
174
Fix-points and Factorization of Meromorphic Functions
IT on the other hand, t = degP, then there exist some arbitrarily small positive number e and e' such that
N (r, ; ) = N (r, ;,) + O(log r) ~
(1 + e)kT(r, F)
~
k(t - 1)(1 + e') T(r, g) .
This will contradict Eq. (4.57) unless t = 1, i.e., P is a linear polynomial. The cases, like P'(s") = 0 can be proposed as having a root, s"o, such that g(z) = S"O has an infinite number of roots, or P'(s") can have at least two distinct zeros and one of them, s"b can enable g(z) = s"l to have an infinite number of roots, etc.; can be argued as before and similar contradictions will result. This also completes the proof of the theorem.
Remark. Noda noted that the condition N (r, j.,) > kT(r, F) of the theorem can be replace by either (i) requiring N (r, j.,) ~ kT(r, F) on a set of r values of infinite measure for some k > 0 or (ii)
N (r, ;, ) - [N (r,
~)
- N (r,
~ ) ] ~ kT (r, ~),
n.e ..
These two facts are useful in the proof of Theorem 4.9 found in the next section. 4.5. THE DISTRIBUTION OF THE PRIME FUNCTIONS We would like to know like the distribution of prime number r in the set of integers; the distribution of prime functions in the family of entire functions. In the section we shall resolve two related questions: (A) (Gross) Given any entire function I, does there exist a polynomial Q such that 1+ Q is prime? B) (Gross, Osgood and Yang) Given any entire function I, does there exist an entire function g such that gl (the product) is prime? Noda provided affirmative answers to the above two questions as follows.
Theorem 4.9. (Noda) Let I(z) be a transcendental entire function. Then the set {ala E CD and I(z) + az is not prime} is at most a countable set in the complex plane 2, either f satisfies the following equation:
where the Adz)(j ~ k ~ n) are rational functions or g must assume one of the forms mentioned in (i).
Proof. We will first prove the case where n = 2. It follows from the assumptions F = f(g) and F" = h(g) that we have
J"(g)g'2 + f'(g)g" = h(g) .
(4.112)
Application of Theorem 4.12 gives the following identity:
A(g)g'2 + B(g)g" + C(g) = 0, where A(z), B(z) and C(z) are polynomials with ABC g,2 from Eqs. (4.112) and (4.113), we get
(4.113)
1:-
O. Eliminating
[A(g)J'(g) - B(g) + J"(g)]g" = A(g)h(g) + f"(g)C(g) . IT A(g)f'(g) - B(g)f"(g)
1:- 0,
(4.114)
then from the above identity,
g" = [A(g)h(g) + J"(g)C(g)]j[A(g)f'(g) - B(g)J"(g)] = Hdg) , where Hdz) is a meromorphic function. Clearly, HI cannot be transcendental. Furthermore, it is easily shown that HI must be a linear function. Hence we have g" = ag + b • (4.115)
198
Fix-points and Factorization of Meromorphic Functions
Substituting this equation into Eq. (4.112), we get
g,2
= [h(g) - !,(g)(ag + b)lI !,,(g) = H2(g) .
Similarly, we can conclude that H2(Z) must be a polynomial of degree Thus
g,2 = t11
~
2.
+ t2g + t3
= tdg - sd(g - S2) , where t. and Sj are constants. It follows, depending on Sl = S2 or Sl that g assumes one of the forms stated in (i). Now we consider the case: A(g)f'(g) - B(g)f"(g) = 0, i.e.,
f"(w) !'(w)
A(w) B(w) .
=I S2,
(4.116)
Two cases will be considered separately: case (a): A(w) is a constant and case (b): A(n) is not a constant. We treat case (a) first. In this case, we may assume without loss of generality that A(w) = 1. Equation (4.113) becomes w,2B(w)w' + C(w) = o. (4.117) Set
B(w) = bW d1 + Bt{w),
C(w) = cw d2 + Ct{w) ,
where d 1, d2 are the degrees of B(w) and C(w) respectively. Then, by Wittich's result on the existence theorem of solutions of certain differential equations, either d 1 + 1 = d 2 > 2 or max(d 1 + 1, d2 ) = 2. Suppose that d 1 + 1 = d 2 > 2, then by rewriting Eq. (4.117) as (4.118) and applying Clunie's result [po 68], we have
T(r, bw" + cw) = m(r, bw" + cw) = s(r, w) .
(4.119)
The central index tJ(r) of g satisfies b(tJjz)2 + c(1 + kt{z)) = k2(Z), where k1(Z) and k2(Z) tend to zero as Izl - 00 (outside possibly a set of
Fix -points and Theory of Factorization
r(=
199
Izl)
values of finite length. Therefore the order of 9 is no greater than 1. However, according to a result of Ngoom and Ostrovskii's, we have
( t:...)
_ m, r'l lim r--+oo log r
= max(t - 1,0)
for any merom orphic function f of order t « 00). Thus the term S(r,g) in the equation has a magnitude 0(1) log rand bg" + cg can only be a constant. This leads to the situation seen in Eq. (4.115) we encountered before. We now treat the situation: max(d 1 + 1,d2 ) = 2. Eq. (4.117) then becomes (4.120) W'2 + (b 0 + blW ) W/I + Co + CIW + C2W2 =- 0 , where b1 and Cl are the leading coefficient of Bdw) and C1(w) respectively. Again by the central index theorem, we derive (4.121) where hdz) and h2(Z) tend to zero as Izl -+ 00, outside a set of r(= values of finite length. In the meantime, we have
Izl)
f"(w) = A(w) = 1 f'(w) B(w) b, w + bo It follows from examining the residue that b1l ¥- 0, -1 and 1. We conclude from Eq. (4.121) that the order of 9 is :5 1. If g' never vanishes, then we are done. If we assume that g'(zo) = 0 for some zo, then be differentiating
W,2 + B(w)w" + C(w) =
0 .
By setting z = Zo, we get,
B(wo)g'''(zo) = 0;
Wo = g(zo) .
°
Two cases may arise: (i) B(wo) = and case (ii) B(wo) ¥- o. If B(wo) = 0, then B(w) = bow. Substituting this result into Eq. (4.11) and letting z = Zo yields Co = 0, and Eq. (4.120) becomes (4.122)
200
Fix-points and Factorization of Merom orphic Functions
Two subcases will be considered (ia) CI = 0 and (ib) substituting y = w' /w into Eqs. (4.122) we have,
CI
i
0_ Under (ia) by
(4.123) We note now that y has a simple pole at z = Zo with residue p; where p is an integer ~ 2. Comparing the coefficients of the term %":%0 in the above equation, we see On the other hand, bI1 = q is also an integer i 0, -1, and 1. Then the above equation yields (q + l)p = 1, which is a contradiction. We now consider case (ib): Since the order of g is no greater than one, we are done if g never vanishes. So we assume g(zd = 0 for some ZI. Then it follows from Eq. (4.122) that g'(ZI) = 0, but gll(ZI) i 0 (since CI i 0). Thus every zero of g is of multiplicity 2. Hence g(z) = K 2(z) for some entire function K and Eq. (4.12) becomes (4.124) Differentiating above equation we get (3b l
+ 4)K'(zI)K"(zI}
= 0.
Since 3b I +4 i 0 (as bI1 is an integer) and K~(zI} i 0 we conclude K~/(ZO) = O. Therefore K" / K is an entire function (since K = 0 has only simple
(r,
(r,
zeros). Thus m ~') = T ~') = o(l)log r. It follows that constant. Substituting this into Eq. (4.124) we get
~'
is a
From g = K2 we have g' = 2K K' and, hence, g'2 = 4K2 K'2 = 4g( d l g+ d2). This goes back to Eq. (4.115). Thus case (i) is settled completely. Now we discuss case (ii): B(wo) i o. Again if g' never vanishes then we are done. Therefore, we assume that gl(ZO) = 0 for some z. Then from Eq. (4.117) we can derive the same conclusion gll(ZO) = 0 (but gll(ZO) i 0 by the " ) uniqueness theorem for the equation w" = - C(w) B(w) - B(w) • In a similar manner, we find that gI' / g is entire and, moreover, it must be a constant. This leads to the form found in Eq. (4.115), that has been
Fix-points and Theory of Factorization
201
settled already. All the above discussions conclude the case where A(w) is a constant. To complete the proof for the case n = 2 we need to settle case (b); that A(w) is not a constant. We may assume that A(O) = 0 and shall treat two subcases separately: sub case (b1) B(O) = 0 and subcase (b2) B(O) =F o. Suppose that case (b1) holds. Then it may also be assumed, without loss of generality, that A, B, C are relatively prime. It follows, from A(O) = B(O) = 0, that C(O) =F o. we recall a result of A.Z. Mokhouko and V.D. Mokhouko. Suppose that P(z, w, ... ,w(n)) is a differential polynomial in w with polynomials as the coefficients and that f is a transcendental, meromorphic function solution of P(z, w, w', . .. ,w(n)) = 0 with P(z, 0,0,0, ... ,0) ~ O. Then m
(r,
7)
= S(r, f) .
Therefore, by applying this result to P(z, w, ... ,w(n)) == A(w) +B(w)w" + C(w) (and noting that P(z, 0,0, ... ,0) = C(O) ~ 0), we get (4.125) On the other hand; from C(O) ~ 0, A(O) = B(O) = 0 it becomes clear that g never vanishes. This contradicts Eq. (4.125). Then case (b1) has to be excluded. We now proceed to settle case (b2): Let Zo be a zero of g(z) with multiplicity t and g"(zo) = -C(O)/B(O) = d (a constant). Then if t = 1,
_ g"(z) K(z) ( ) gz
d(_= w" - d) w
,
(4.126)
will be regular at z = zoo We are going to show that it is impossible to have t ~ 2. Otherwise, from Eq. (4.115), we can successively derive g'(zo) = gn(zo) = ... = g(n)(zo) = 0, '
~logM(~,fog)
10gM(r,f)
>
~logM(~,g) +0(1) --+ 00 logr asr--+oo.
and obtain the desired result. To prove Eq. (11) we need one of Clunies' earlier results. IT 1 and 9 are entire, then,
M(r,/o g)
~
M((1- 0(1))M(r, g), f),
r --+ 00,
r
t- E ,
where E is a set of r of finite measure and 0(1) and E are depending on g. As I(z) is transcendental, the above inequality implies that, for any given positive constant k, since M(r,1 0 g) ~ (1 - a(1))k(M(r, g))k as r --+ 00, r E, -1' log M(r, log) k 1m r-+oo log M ( r, ) 9 > - ,
t-
and k can be chosen to be arbitrarily large, the result follows.
212
Appendix
To prove Eq. (12), we recall a well known fact in the Nevanlinna valuedistribution theory; namely if J is meromorphic then for all complex number w outside a set of zero capacity depending on J,
It follows from this result, that a constant a can be chosen so that J(z) - a has an infinite number of zeros, ~t. ~2,' •• ,~n, ... and
N (r, J
1
og- a
) "" T(r, Jog)
N (r, _1_) "" T(r,g) g-
as r
~n
as r
-+ 00 ,
-+ 00,
n = 1,2, ...
It follows that, for any given positive integer n,
N
(
r,
1) > ~ (1) r,-~i n
Jog - a
and so
-
N
T(r, Jog) >
1.
1m
r-+oo
) T( r, 9
-
9-
n.
As n can be chosen to be arbitrarily large the result in Eq. (12) follows. Theorem A.S. (Clunie) Let J(z) be meromorphic and g(z) be entire and suppose that J(z) and g(z) are transcendental. Then
-1' T(r,Jog)_
r!.~
T(r, J)
-
00 .
Theorem A.6. (Clunie) (i) Let J(z) be transcendental meromorphic and 9 be transcendental entire. Suppose that at least one of them is of finite order. Then 1. T(r, Jog) _ 1m r-+oo
(ii) Let Then
J and
T( r, J)
-
00 .
9 be given as in (i). Suppose that
r
r!.~
log M(r, Jog) log M(r, g)
=
00 •
g(z) is of finite order.
213
Appendix
Remarks. (1) By constructing an example Clunie showed that the finiteness of the order of 9 is a necessary condition for the validity of (ii) of Theorem A.6. (2) Clunie also demonstrated (by example) that for a certain pair of functions 1 (meromorphic) and 9 (entire). lim T(r, log) r~oo
=0
.
T(r, f)
(3) Adopting Clunie's reasoning, Song and Yang showed (i) there exists an entire function 9 such that . log log M(r, eg ) 11m r~oo log M(r, g)
=0
,
and (ii) there exists a meromorphic function such that lim logT(r,1 0 g) = 0 . r~oo log T(r, g)
1
and entire function 9
3. THE EXTENSION OF POLYA'S THEOREM TO MEROMORPHIC FUNCTIONS
Theorem A.'T. (Edrei and Fuchs) Let I(z) be a meromorphic function that is not of zero order and 9 be a transcendental entire function. Then I(g) is of infinite order. The above is an extension of Polya's theorem and is an immediate consequence of the following result.
Theorem A.S. (Edrei and Fuchs) Let 1 and 9 be entire functions. Assume that the zeros of 1 have a positive exponent of convergence and that 9 is transcendental. Then the zeros of I(g) do not have a finite exponent of convergence. 4. SOME NECESSARY CONDITIONS FOR THE EXISTENCE OF MEROMORPHIC SOLUTIONS OF CERTAIN DIFFERENTIAL EQUATIONS
Theorem A.9. (Steinmetz, Gackstatter and Laine) Let
P(z,w,w', ... ,w(n)) == LO!i(Z)WiO(w')i1 ... (w(n))in. AEI
Appendix
214
be a differential polynomial in w(z) with the coefficients aj(z) being nonzero meromorphic functions, where I is a finite set of multi-indices >. = (io,i 1 , ... ,in) (io,i ll ... ,in are nonnegative integers). Let q
p
A(z, w) =
L
B(z, w)
aj(z)w(z)j,
=
L
bk(z)w(z)k ,
k=O
j=O
where aj(z) and bk(z) are nonzero meromorphic functions with ap(z)bq(z) ~
o. Consider the differential equation
P(z, w, ... ,w(n))
=
A(z, w)j B(z, w)
and set ~
= max(i o + >'EI
2il
+ ... + (n + l)i n )
,
d = max(io + i 1 + ... + in) , >'EI
and
If the above differential equation has a meromorphic solution w(z) satisfying
T(r, c(z)) = oT(r,w)
as
r
--+ 00 ,
outside a set of finite measure Ej where c(z) represents any of the coefficients (i.e., Cj, ai, bk ) in the equation, then (i) q = 0 and p ~ ~ and (ii) p ~ d under the additional condition that N(r, w) oT(r, w) as r --+ 00 outside a set of finite measure.
5. SOME PROPERTIES OF DIFFERENTIAL POLYNOMIALS Let M define the class of all the meromorphic function. We shall represent as S(r, I) any quantity satisfying S(r, I) = o{T(r, as r --+ 00, possibly outside 'a set of r' of finite measure. Let P(z, I) be a polynomial in I and its derivatives with the coefficient a(z) satisfying T(r, a(z)) = S(r, I). We shall call P(z, I)) a differential
In
215
Appendix
polynomial in I (or simply a differential polynomial P(J)) and Pn(z, I) denotes differential polynomial of degree at most n in I.
Theorem A.IO. (Clunie) Let tions. Suppose that
I
be a transcendental meromorphic func-
where P(z, f) and Qm(z, f) are both differential polynomials in I with m ~ n. Then
m(r, P(z, I)) = S(r, f) . The above result and the one below are both contained in Hayman's book Meromorphic Functions.
Theorem A.II. (Tumura and Clunie) Let I(z) be a non-constant meromorphic function. Suppose that
g(z) = I(zt
+ Pn-df) ,
and that
= h(z)n, h(z) = I(z) + ~a(z), and a(z) is obtained by equating h(z)n-l (z)a(z) with the terms of degree n-l in Pn-df) after substituting h(z) for I(z), h'(z) for J'(z), etc. For example, if Pn-df) = ao(z)/'(z)/(z)n-2 + Pn- 2(J), then Then g(z)
and hence
h' ao(z) g' a(z) = ao(z)- = - - - . h
n
g
Therefore, in this case
g(z) = hn(z) = (/(Z)
+ ao (z) g' (z)) n n
g(z)
Appendix
216
6. A SIMPLER PROOF OF STEINMETZ'S THEOREM The following is a simpler proof of Steinmetz's Theorem due to GrossOsgood. The method is motivated by some techniques employed in the study of transcendental number theory.
Theorem A.12. (Steinmetz) Suppose g is entire, n ~ 2 is a natural number, and fdz) 1- 0(1 ~ i ~ n) and hi(z) 1- 0 are meromorphic. Suppose n
that
E
i=l
T(r,hi) = O(1)T(r, g). If
n
E
i=l
Ji(g)hi(z) == 0, then there exist n
polynomials, Pdz), not all zero such that E~=l Pi(g)hi(z)
= o.
Lemma. Let F1 1- 0, F2 1- 0, ... ,Fm 1- 0 be m formal power series in Z - a for any complex number a. Then there exists an infinite sequence of (m + 1)-tuples of polynomials in z, (Qi' P1i(z), P2i (Z), ... , Pmi(z)) that satisfy, for each j, the following three properties:
(i) Qi(z) 1- 0 (ii) max{deg Qi' deg Pii , ... ,deg Pmi } ~ mj ,and (iii)
Z
= a is a zero of multiplicity at least
Pii(z), 1 ~ i
(m+1)j for every Qi(z)Fdz)-
~ m.
Proof of the lemma. Property (iii) actually imposes m(m + 1) linear homogeneous conditions on the (yet to be determined) coefficients of Qi(z) and the Pii(z). By (ii) there are no more than (m + 1)(mj + 1) such coefficients to be determined. Since (m+ 1)(mj + 1) > m(m+ 1)j, it follows from the theory of system of linear equations that for each j, there exists a set of coefficients for the Qi(Z) and the Pii(Z) that are not all identically zero, such that (iii) holds. Next we show that Qi(Z) 1- o. Otherwise, we conclude from Pii(z) = Qi(z)Fi(Z) - Pii(Z) and (iii) that each Pii(Z) would vanish at Z = a to an order greater than deg Pii , which would yield Pii == (z), 1 ~ i ~ m and Qi(z) == 0, a contradiction. Proof of the theorem. Set m = n - 1 in the lemma and Fdz) = ~ m. Let a be any point such that each Fi can be expanded
'i:tlj) ,1 ~ i
n-1
into a power series. Define Gi(z) == Qi(g)hdz) +
E
i=l
Pii(g)hHdz), 1
~
j ~ 00. We are going to show that at most a finite number of the Gi(z) are nonzero functions. In what follows we may assume each Gi(z) 1- o. This
217
Appendix
will lead to a contradiction. First we will show that n-l
H .(z) = J
-
Qj(g)h1(z) Gj(z) Ig(z) - al nj
+E
Pij(g)hH1(Z)
i=l
has exactly the same poles as does Gj(z). From hypothesis n
h(g)hi(z) == 0 ,
L
(13)
i=l
we have, by multiplying (13) by Qj(g)/ II (g),
Hj(z) = - [g(z) -
ar nj [Qj(g)hl (z) + ~ Qj(g) f~:(~)) hi+dz)
-(Qj(g)hdz) +
~ Pij(g)hH1(Z))]
.
Thus n-l
Hj(z) = L(Qj(g)Fi(g) - Pij(g))(g(z) - a)-njhHdz) .
(14)
i=l
By (iii) of the lemma, for each j,
is entire, so the division of Gj(z) by [g(z) - ajnj yields no new poles (since g(z) - a is entire, the division cannot remove any pole). Hence n
N(r, Hj(z)) = N(r, Gj(z)) ~
L i=l
Let
T{z:
Ilg(z) - al
Thus for all z in T, we can see that
~ 1} .
N(r, hi)
(15)
Appendix
218
is bounded. Hence, by virtue of (14), we have, for all z E T,
On the other hand, by property (iii) of the lemma, for all z E CC\T. (CC denotes the complex plane), we have for each i and j
I I(g(z)Qi(g) - a)ni
an
d
I(g(z)P'i(g) I _ a)ni
are bounded. Hence,
Thus, we have
n
m(r, Hi(z)) ~
L m(r, h.) + k
(16)
.=1 where k is a positive constant independent of r. It follows from (14) and (15) that n
T(r, Hi(z)) ~ LT(r, h.)
+k=
O(I)T(r, g)
+k .
(17)
.=1 Next we proceed to estimate m
(r, ~j) which by Nevanlinna's first funda-
mental theorem will be no larger than the right hand side of (17) (possibly for a new constant replacing k). We denote this bound by B i . If j is suffi-
(r,
ciently large, we shall derive a lower bound for m ~j) which will exceed Bi and the theorem will then be proved. For all z E CC\T, we have from (ii) of the lemma that
IHi(z)1 ~ O(I)(lg(z) - al)-i max{lh.(z)l; 1 ~ i ~ m} . Since the m,ax deg {Qi' P'i} ~ (n-l)j. Thus, for some positive constant l:s;.:S;m
d, independent of z,
I
log+ H i 1(Z)
I ~ log IH3~(Z) I ~ jlog Ig(z) -
n
- L log+ Ih.(z)l- d = jlog+ Ig(z) .=1
al .
n
al-
L log+ Ih.(z)l- d. .=1
(18)
219
Appendix
(Note that (18) holds for all z ETas well.) We obtain by averaging (18) over the circle: m (r,
Izl = r,
Hj~Z)) ~ iT(r, g) - ~ T(r, hd -
d ::; B j .
The above inequality is impossible to hold for sufficiently large completes the proof of the theorem
i.
This also
REFERENCES
1. I.N. Baker and F. Gross, "Further results on factorization of entire functions", Proc. Symposia Pure Math. Amer. Math. Soc., Providence, R.I. II, (1968) 30-35. 2. J. Clunie, "The composition of entire and meromorphic functions", McIntyre Memorial Volume, Ohio Univ. Press (1970). 3. A. Edrei and W.H.J. Fuchs, "Sur les valeurs deficientes et les valeurs asymptotiques des fonctions meromorphes", Comment. Math. He/v. 33 (1959) 258-295. 4. A. Edrei and W.H.J. Fuchs, "On the zeros of f(g(z)) where f and g are entire functions", J. Analyse Math. 12 (1964) 243. 5. R. Goldstein, "On factorization of certain entire functions", J. London Math. Soc., (2) (1970) 221-224. 6. R. Goldstein, "On factorization of certain entire functions, II", Proc. London Math. Soc. 22 (1971) 483-506. 7. F. Gross and C.C. Yang, "Further results on prime entire functions", Trans. Amer. Math. Soc. 142 (1974) 347-355. 8. F. Gross, Factorization of Meromorphic Functions, U.S. Government printing office, Washington, D.C. (1972). 9. W.K. Hayman, Meromorphic Functions, Oxford Univ. Press, Oxford (1964). 10. M. Ozawa, "On prime entire functions, I and II", Kodai Math. Sem. Rep. 22 (1975) 301-308, 309-312. 11. M. Ozawa, "Sufficient conditions for an entire function to be pseudoprime", Kodai Math. Sem. Rep. 27 (1976) 373-378. 12. M. Ozawa, "On uniquely factorizable meromorphic functions", Kodai Math. J. 1 (1978) 339-353. 221
222
References
13. G.S. Prokopovich, "On superposition of some entire functions", Ukrain. Mat. Zh. 26, No.2, March-April (1974) 188-195. 14. G.S. Prokopovich, "On pseudo-simplicity of some meromorphic functions", Ukrain. Mat. Zh. 21, No.2, March-April (1975) 261-273. 15. G.S. Prokopovich, "Fix-points of merom orphic functions", Ukrain. Mat. Zh. 25, No.2 (1972) 248-260 (English translation 198-208). 16. J.F. Ritt, "Prime and composition polynomials", Trans. Amer. Math. Soc. 23 (1922). 17. P.C. Rosenbloom, "The fix-points of entire functions", Medd. Lunds Univ. Mat. Sem., Suppl. Bd. M. Riesz (1952) 186-192. 18. H. Selberg, "Algebroid functions and inverse functions of abelian integrals", Arhandlinger utgittav det norske Videnskaps-Akademi i Oslo I. Matem.-Naturvid. 8 (1934) 1-72. 19. G.D. Song and C.C. Yang, "On pseudo-primality of the combination of meromorphic functions satisfying linear differential equations, in value distribution theory and its applications" , edited by C.C. Yang, Contemporary Math-series 25, American Math. Soc. Providence, R.I. (1980). 20. N. Steinmetz, "Uber die fakorisierbaren Losungen gewohnlichen Differentialgleichungen", Math. Zeit. 110 (1980) 169-180. 21. N. Toda, "On the growth of merom orphic solutions of an algebraic differential equations", Proc. Japan Acad. 60, Ser. A (1984) 117-120. 22. H. Urabe, "Uniqueness of the factorization under composition of certain entire functions", J. of Math. of Kyoto University 18, No.1 (1978). 23. H. Wittich, Neuere Untersuchungen tiber Eindeutige Analytische Funktionen, Springer-Verlag, New York (1984). 24. H. Wittich, "Ganze transendente Losungen algebraischen differentialgleichungen", Math. Ann. 122 (1950). 25. C. Yang, Factorization Theory of Meromorphic Functions, Lecture Notes in Pure and Applied Mathematics, Vol. 18 (edited by C. Yang), Marcel Dekker, Inc (1983).
INDEX
algebroidal function, 155, 156 Baker, LN., 70, 109 Bergweiler, W., 149, 210 Bohr, H., 59 Borel, 24, 53 exceptional value, 34 Borel's lemma, 121, 123 theorem for meromorphic function, 34 Brownawell, w.n., 183 Bureau, F., 53 Cartan, H., 8 Cauchy inequalities, 14 characteristic function, 6 Chebyshev polynomial, 120, 128 Chuang, Chi-Tai, 1 Clunie, J., 140, 170,207,211,212,213,215 Clunie's theorem, 136 completely invariant, 87, 99, 100 completely multiple value, 36 conjecture 1, 150, 192, 193 2, 171 9, 180 ..I, 192
223
224
Index
convex function, 9 deficiency, 36 deficient value, 36 diagonal sequence, 82 differential equation solutions of, 181 algebraic, 187 differential operator, 189 differential polynomial, 203, 214, 215 Edrei, A., 119 Edrei-Fuchs, 168, 170, 213 elliptic function, 134, 136 factorization of, 134 entire function, 2 exponential type, 116 systems of, 38 transcendental, 14, 36, 37 E-prime definition of, 109 E-pseudo-prime definition of, 109 equivalent factorization definition of, 146 exponent of convergence, 213 factorizability, 108 factorization, 108 definition of, 109 Fatou, P., 107 Fatou's theorem, 94 theory on the fix-points of entire function, 85 finite positive order, 34 fix-point, 65 attractive, 88 definition of, 107 neutral, 91 repulsive, 89 Fuchs' conjecture, 171
Index
functional equations, 139 functional identities, 205 functions in cosine or exponential forms factorization of, 119 fundamental theorem first, 12 second, 28 gamma function, 188 Goldberg, A.A., 109 Goldstein, R., 109, 139, 145, 150, 154, 162, 167, 170 Gross, F., 109, 118, 148, 149, 150, 162, 174, 187, 197, 210 Gross' conjecture, 150, 181 Gross-Osgood, 187, 193, 216 Gross-Yang, 162, 171 growth,13 growths of f(g), f and g, 206 Hayman, W.K., 1 holomorphic functions normal families of, 76 identity, 38 irreducible polynomial, 126 Jensen formula, 4 Jensen-Nevanlinna formula, 6 Julia point, 85 set, 85 set of polynomials and rational functions, 105 left factor definition of, 109 left-prime definition of, 171 linear transformation, 146 local uniform convergence, 76 locally uniformly bounded, 76 logarithmic derivative, 16 meromorphic function, 2
226
Index
fix-points of, 49 linearly independent, 38 Nevanlinna's theorem of, 49 systems of, 38 transcendental, 14, 33 Milloux, H., 53 minimum modulus, 164 Mokhouko, A.Z., 201 Mokhouko, V.D., 201 Montel's theory of normal families, 49 multiple points, 32 Nevanlinna exceptional value, 36 Nevanlinna, R., 1 Noda, Y., 109, 174, 177 non-decreasing function, 9 non-periodic functions factorizing, 117 Osgood, C.F., 174, 187 Ozawa, M., 109, 118, 119, 123, 124, 171, 172, 191, 192, 205 periodic entire function, 119 periodic mod g definition of, 110 Picard exceptional value, 33, 151, 157 Picard's theorem, 107, 126 theorem for meromorphic functions, 33 point of accumulation, 76, 94 Poisson-Jensen formula, 2 P6lya, G., 60, 210, 213 P6lya's theorem, 121, 135, 151, 193, 208 polynomials, 100 prime, 108 definition of, 109 Prokopovich, G.S., 109, 150, 154, 156, 162 pseudo-prime, 108 definition of, 109 Riemann surface, 126 right-factor
Index
definition of, 109 common, 196 right-prime definition of, 171 Ritt, J.F., 110,127,146 Rosenbloom, P.C., 65, 107, 108 Schwarz lemma, 77 Selberg, H., 155, 156 small function, 37 Song, G.D., 188, 210 Steinmetz, N., 109, 182, 188, 196, 202, 213 Steinmetz's theorem, 182, 216 Steinmetz-Yang, 197 Toda, N., 206 transcendental number theory, 216 Tumura-Clunie, 215 Tumura-Clunie theorem, 147, 167 uniquely factorizable definition of, 146 uniqueness of factorization, 146 Urabe, H., 109, 118, 147, 191 Vitali, G., theorem of, 79 Wiman-Valiron theorem, 149 Wittich, H., 182, 191 Yang, C.C., 109,118,150,162,174,188,191,193 Yang-Gross, 119