VOLUME
5
B.Ja. Levin
Distribution of Zeros of Entire Functions Revised Edition
American Mathematical Society
Translations
of Mathematical Monographs
Distribution
Volume 5
of
ZEROS OF ENTIRE FUNCTIONS Revised Edition
by B.
Ja. Levin
AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND
PACTIPEJlEJIEHHE KOPHEA UEJIhIX YHKUHA E. H. JIEBHH rocy.napcrBeHHoe H3J(8Te.jILCTBO
TexHHIr
and
lim cp (r) r-+ 0 for
and therefore asymptotically
(1.09)
Sec. 2]
TAYLOR COEFFICIENTS
5
Conversely, assume that (1.09) holds for all indices n greater than no(k, A), and let us estimate M,{r). For n > mr = [2 k eAkrk] and all sufficiently large values of r we have by (1.09) and therefore
mr
If(z) I
< ~ Ic"lr"+2- mr • ,,:0
Introducing the notation Il-(r) = max I c" Ir",
"
we have
-< (1 + 2keAkrk) IJ. (r)+ 2- tnr.
(1.10) If j(z) is not a polynomial, then M,(r), and therefore by (\.10) also /J(r), grows faster than any power of r, and therefore the index of the largest term in the series (1.03) increases without bound as r grows. It follows from (1.09) that asymptotically M,(r)
The maximum of the right side is attained for n = Akr k
and therefore asymptotically
-
0 and all complex numbers u
a
•
In I (a, p) I
where
< API1ul+ I+u I ' p
1
(1.21)
Ap= 3e(2+ Inp).
Eorp = 0
In Ia (a; 0) 1-< In (1 + Ia I). The second assertion of the lemma is obvious. Let p O.
If
uP } InIO(a; p)I=Re { In(l-a)+a+ 2u3 + .. , +p
~
u"
~ I u"I"
I u IP+l
,,+1
= -Re ~T-< ~-k- < (p+ 1)(1-lul) - p/(P + 1) the inequality In (1 + luI) .,
r
L,
1P
+ P + ~ _ A] ,
that is, the order 4 of II(z) does not exceed A, and therefore, it does not exceed Pl' If PI = P + 1, then, as was shown in the proof of Lemma 1, lim
t~co
~=O 1 tP +
and the integral <Xl
f o
n (t) dt
tP + 2
converges. Using this we have from the lemma that asymptotically In I II (z) I
that is, II(z) is at most of order P REMARK.
< erp + 1 ,
+ 1 and minimal type.
If PI is not an integer and if the upper density of the sequence
{an} is finite, then II(z) is at most of order PI and normal type; if PI is not an integer and if the density ~ of the sequence {an} is zero, then n(z) is at most of order PI and minimal type. This assertion follows easily from the asymptotic inequality n(t)
< (.l+e)tP.,
which, for PI not an integer, leads to the asymptotic inequality In I II (z) I
< (1 +e) Bp,rP ••
(1.25)
Note that we cannot obtain this inequality from Lemma 3 in case PI is an integer. Later on we shall see that for integral PI the assertion ceases to be true. For the further investigation of the connection between the growth of the function n(r) and the growth of In Mn(r) we require a series of general theorems that playa fundamental role in the theory of entire functions. • Clearly in this inequality one can replace 10(z)1 by Mn(r).
14
GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS
[CHAP. I
5. Jenseo's Theorem THEOREM 5. Let fez) be holomorphic in a circle of radius R with center at the origin, and f(O) ~ O. Then R
2_
J
J IJ(Re'8) Id~
2~
n't(t) dt =
In
o
-In 11(0)1,
(1.26)
0
Izl < t. the.circle Izl < R,
where n,(t) is the number of zeros offez) in the circle
Note that if the function has no zeros in then net) = 0 and equation (1.26) expresses a well-known property of harmonic functions. If there are zeros, then it follows from (1.26) that
..
2~
In 1/(0) 1
0 satisfies, for 1m z and Izl > 1, the inequalities
~ I/(i)1 51
;0 < I f(z) I 0 >0
(1.31)
< 'It).
This inequality will be called the CaratModory inequality for the half-plane. PROOF.
Map the upper half-plane 1m z
z-i u = -z+i
or
Z
> 0 onto the unit circle
=-
a+l a-I
1--.
The function
F (u) = 11(- i a-I a + 1) is defined in the unit circle and satisfies Re F(u) -H(Tj) In M (2eR) H (Tj) = 2
PROOF.
Q2"
+ In 3e2-tj'
Construct the function
cp (z) = where a l ,
(1.34)
(-2R)" altZ:i .•• a"
" 2R (z-ale) II , Ie
=1
(2R)2 -
an are the roots ofj.(z) in the circle
•• ,
cp (0) = 1 and Icp (2Re i8 ) I =
De function IJI(Z)
K
alez
Izl < 2R. We have
(2R)"
I altZ:i ••• a" I
•
= f(z)
,00
_ no roots in the circle Izl ~ 2R, and therefore by the corollary to the Caratheodory theorem for the circle (Theorem 9) we have for Izl ~ R In
IHz) I> -
>-
21n M,(2eR)+ 21n i al~2~)."a" I 21n M,(2eR).
Now we estimate I!p(z) I from below. For Izl :( R
II" I (2R)9 -
a"z I < (6R'l)fI •
(1.35)
22
M,(R1) > e( "-2"• ) B1P > e("-2" \ - ) "1.
Comparing this with (1.37) we have
In M,.(r1)> [(0- ; )(l-Wr~ -(2+ In 1~) In M,(2eR)].
(1.37)
24
GENERAL THEORY OF THE GROWTH OF ENTIRE fUNCTIONS
[CHAP. I
and since we have
In M,,(r1)> [(0- ;)(1-3>'-(2+ In 1~)(1-o)-P(2e>,o]r~. For given E > 0 one can choose 6 so small that the expression in the square brackets is not less than (1 - E. Consequently,
M,,(r 1)
> e("-·l~
for a sequence of values r 1 tending to infinity, and the theorem is proved. COROLLARY. If the quotient of two entire functions f(z) and !p(z) is an entire function lp(z), then its category does not exceed the larger of the categories of the functions fez) and !p{z); here the categories of fez) and !p{z) may be the same. If they have different categories, then the category of the function lp(z) equals the larger of the categories offez) and !p{z). PROOF.
We are given
cp (z) ~ (z) = ! (z). If the category of lp(z) is larger than the category of !p{z), then the category off(z) is equal to the category of lJ'{z). Thus the category of lp(z) cannot exceed the categories of fez) and !p(z), and the first part of the corollary is proved. Clearly also, if the category off(z) is larger than the category of !p{z), then the categories of fez) and lp(z) are equal. On the other hand, the category of fez) can be smaller than the category of !p(z) only in case the categories of !p{z) and 1p{z) are equal. 10. Hadamard's Theorem
The theorems of the preceding sections enable us to refine considerably the theorem on the representation of an entire function as an infinite product. This refinement, which is due to Hadamard, concerns the representation of entire functions of finite order, and is one of the classical theorems of the theory of entire functions. THEOREM
13. The entire function f (z) offinite order p can be represented in
the form
•
!(z)=z"'eP(.1
II a (.!....; p) lin
«(1)-
PROOF.
=
Consider first the case p fR (z) =
p. We introduce the function
IT a (a:; P-
1)
II a (a:; p),
( 1.39)
I an I> R
I an I" R
with the aid ofwhichf(z) can be represented in the form
() f( z-ze ) - m Pp-l(') e"f(R),P f RZ'
( 1.40)
where P,,-1(:::) is a polynomial of degree at most p - I. T~is important representation will be used frequently in the study of entire functions of integral order. To estimate the growth of the function(u(:::) we use inequalities (1.21) and (1.22). Assume that p > I. Then, if we put MR (r) =
max
IfR (rei~) I.
0.,;;:8 atl' or more generally if p' == p, • ~ == atl' ••• , 1X1 -1 == at;_l but IX} > atl' All the theorems of the preceding llections can be generalized to this more precise growth scale. In particular, it can be shown that if p is not an integer, then the asymptotic inequalities
nr(') ~
< ,P In'" T
•••
In;II+o'
equivalent, and therefore in this case the generalized orders of the functions lItCr) and n,(r) are equal. . We shall not prove these theorems. as we shall soon prove more general . theorems. The growth scale can be further refined by introducing as comparison
32
GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS
[CHAP. I
functions monotone functions that grow more slowly than any of the iterated logarithms In,r, etc. However, instead of such further refinements, it is more natural to define in a general manner a certain class of "slowly increasing" functions L(r) and then to compare In M,(r) with functions of the form rPL(r). This was the path taken by Valiron [1], who introduced the concept of proximate order of growth. A function p(r) that satisfies the conditions l l lim p(r)
= p ;;> 0
and lim rp' (r) In r
=0
(1.52)
is called a proximate order. If for the entire function f(z) the quantity - . InM,(r) 0,= , lim ( ... co r P r) is different from zero and infinity, then p(r) is called a proximate order of the given entire function f(z), and at is called the type of the function f(z) with respect to the proximate order p(r). Clearly, the proximate order and the corresponding type of the given function are not uniquely determined. For example, if we add In clln r to the proximate order, then we obtain a new proximate order for the given function, and now the type has been divided by c. With respect to an arbitrary proximate order, a function may be of minimal, normal, or maximal type. With respect to its own proximate order it is always of normal type. A positive function will be called slowly increasing, and will be denoted by L(r), if lim L (kr) = 1 , ... co
L (r)
uniformly on each interval 0 < a ~ k 0( b < 00. The following lemma plays a fundamental role in the study of proximate orders. LEMMA
5.
If p(r) is a proximate order, then the function r P(,)-p
is slowly increasing. 12 PROOF.
then
Let L (kr)
In L(r) =(p(kr)-p)lnk+(p(kr)-p(r»)lnr. II Here and throughout this book the assertion lim cp(r) = A means "the limit exists and is equal to A." 11 The converse is true in the presence of certain supplementary restrictions on the function 1.(r) (for example. if 1.(r) is logarithmically convex).
Sec. 12]
33
PROXIMATE ORDERS
To fix ideas let us assume that 0 < (/ .;;; k < I. Then by the second of conditions (1.52), for arbitrary 1] > 0 and all sufficiently large values of r we have by Lagrange's theorem:
Ip(r) -
p (kr) I
0).
where A __
~
Indeed. for
E
> 0 and r > r. 'P(t)
-I.
,,(r)
r .. co
rf (r)
1m - - .
< (A+a)tP(t)
and consequently r
r
S";t) dt ~O(l)+(K+a) Sfl'l-).dt. •
r.
+1
Sec. 12]
35
PROXIMATE ORDERS
Applying Rule (3), we obtain
f ,( 1"
t)
-
t).
-
dt ~ ~
(11
rP + I) p+l-A. + 0 (rP (1")+1-).
(1")+1-).).
II
From this we have at once
lim 1"-ho
{r-
r
p (1")-1+A
f ,~) t
dt}
-< p+A.-l ~ .
II
Similarly one sees that
lim
-
r
{r- p
(1")-1+).S
'(t)dt}........
A -?'p+l-A.'
f
•
r~oo
where A = Urn ,(r) •
-
-
r+oo
rPl1" )
From Rule (4) and the last remark we have at once the following assertion: (5) If cp{t) is a bounded function on each finite interval and if the limit
11 = lim ,(r) r+oo rP (1")
exists, then for .A.
0).
(1.55)
Also, by properties (d) and (e), there exists a sequence of values r .. tending to infinity for which equality holds in (1.55). It remains to verify that the function p(r) that we have constructed is a proximate order, that is, that it satisfies the conditions (1.52) of Valiron. Clearly, from (b) it follows that limr-cop(r) - p. Also, ~
(In r)
rp'(r)lnr =~'(Inr)- -Inr -. Hence, from (b) and (1.54), we see that limr..... aorp'(r) In r - O. We constructed the function per) under the condition that lim sUPZ--+co9'l(X) - 00. 3. We now show how the general case can be reduced .to this case. For this purpose we construct a .concave function y - 'Pl(X) so that
lim ~l (x) a: -+ 00
= 0,
lim 1\I~(x) = 0
a:-+oo·
X
and so that lim sUPz--co[9'l{X) + 'Pl(X)] = 00. We construct the curve 'P1(x) in the following manner: pass a segment dl of the line ~"'-".¥
from the origin to the point Xl (Fig. 2) at which
CP1 (x t ) Choose a positive number EI t1a of the line
>-
'1 X 1
+ 1.
< El and from the point (Xl' -E1Xl ) pass a segment
Y+11 Xt = - li (X-X1)
to a point XI
> Xl' at which
38
GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS
[CHAP. I
From the point (x., -E1Xl - Ea(X z - Xl» pass a segment d3 having slope -E3 (0 < E3 < E.) etc. y
r,
-R~~T---~~------~-~
o
r
: \y=rp,(r) '--"-~~-~--y=~(r)
Fig. 2
E3
The positive numbers El , E., E3' •.. , E.. , ••• will be chosen so that El > E. > > ... > E.. > ... and t: .. ~ 0, and the points Xl. Xa, ... , x" • ••• are chosen
so that x" -+- 00. Clearly the polygonal function y structed satisfies the condition
= 1Jit(X) that we have con-
~1 (x) = O.
lim
III -++00 • X
By changing the function tPl(X) in an inessential manner in a neighborhood of each angular point, we can make it everywhere differentiable. Let tpl(X) denote the function with the opposite sign. This function y = tpl(X) has the required properties. Now construct a convex majorant tpa(x) for the function IPl(X) + tpl(X), just as in § 2. Then putting ~ (x)
we obtain
=- ~g (x) - '\"1 (x).
'1
(x) .(: .~ (x)
and, on some sequence {x~} of extreme points tending to infinity, we have
'Pi (x~) = ~ (x~) (n = 1. 2.... ). Also lim oV (x) = 0 and lim III -++00
III -+ +00
0/ (x) = O.
(1.56)
x
Putting, just as before, (Inr) ( ) + 1\11iiT'
p r =p
we obtain from (1.56) lim p(r)=p. r-+oo
11m rp' (r) In r r-++oo
=
lim r-++oo
[.V (In r) _
that is, p(r) is a proximate order. Also fer)
~ rP(r)
r)] =
1\1 (lIn nr
0,
Sec. 12]
39
PROXIMATE ORDERS
and for some sequence r" tending to infinity I (rfI) = r~ (,.fI) • The theorem is proved. Note that the function exp (tp(ln r» is a slowly increasing function L(r). In Chapter II we shall need the following remark. REMARK. The slowly increasing function that we have constructed can be represented, as is easily seen, in the form
L (r) =
,"'.(1:1 ,.)-+.
(III ,.),
where tpl(X) and tpa(x) are unbounded increasing concave functions satisfying condition (c). We shall show that in the expression for L(r) the functions tpl(X) and tp.(x) can be replaced by two other concave functions t'JI(x) and t'J'I,(x) that also satisfy the conditions (a)
Jim 4). (x) =
00,
e-++co
(b)
11m
&t (x)
11-++00
x
= 0,
(I
=
I, 2)
and that satisfy, in addition, the condition
(b')
&r = o.
lim (x) al-+ +co &i (x)
To prove this we make the following construction. Let y = tp(x) be a concave function satisfying the conditions Iims -++ ao tp(x) = 00, Iimz -++ ao tp(x)/x = 0, and let (10) be a line of supportU of this curve. On the line (10) choose a point (xo, Yo) and consider the curve (II)
y
= C~l)+C~l)(X_ x o) - c~1)e-·1 (al-IIIo>,
that is tangent to the line (10) at the point (xo, Yo). In the equation for this curve, £1 is a positive number, and the coefficients c~1), 41) are determined by ~e condition that (10) and (/J be tangent at the point (Xo, yo>
C~l) = .!. (y~ _ C~l~. al
C~l) = Yo
+ .!.. ~ _ C~l~. al
where y~ is the slope of the line (10). The parameter c\l) will be chosen to be positive, but less than y~. Then the coefficient 41) will also be positive. The curve (/J is clearly concave and approaches asymptotically the line y = 41) + c\l)(X - xo). It A line of support of a given curve is a line having points in common with the curve, but .tach that the entire curve lies in one of the closed half-planes determined by the line. (For
.6Irther details on lines of support, see § 19.)
40
GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS
[CHAP. I
Also, from the equation
1" (x) y' (x)
=-
+ c~t)e"
C~t)ll
IID-:I:o)
it follows that on the whole curve (/1)
I I< y" (x)
y' (x)
(1.57)
21 •
If the abscissa Xo is sufficiently large, then that part of the curve 11 that lies to the right of Xo is above the curve Y = tp(x). Choosing the curve (/1) in this manner and then decreasing ell!) while keeping the point (xo, Yo) and the quantity El fixed. we can cause this curve to touch the curve y = tp(x) from above (Fig. 3). Since the curve (/1) is concave and contains no line segments, the point of contact (x~. y~) must be an extreme point of the curve y = tp(x). Now choose numbers E2 < E} and ci2) < cP) and choose a point (Xl' Y1) on the curve (/t) far enough out so that the part of the curve (/2) y = C~2)
+- ci2)(X _
Xl) _
c~2;C-'" (.lI-.lI,),
(this curve is tangent to the curve (/1) at the point (Xl' Y1»' lying to the right of this point, lies above the curve Y = tp(x). Without changing the point (Xl' Y1) or the parameter t 2• we decrease the parameter C~2) until the curve (/2) has a point of contact with the curve Y = tp(x). Next we choose numbers E3 and ci3 ) < c\2)
(!.J
Fig. 3
and a point (XI' yJ on I., and form a curve (/J, etc. This process is continued indefinitely, and the numbers E1 > Ell > E3 > ... , cil) > c\1I) > ciS) > ... are chosen so that E" -+ 0 and cl") -+ O. Now we form a smooth concave curve y = ,<x) from the segments of the curves (10)' (/J, ... , (/,,), ... taken between the points of contact. Clearly lim
:z:-+ + co
0 (x) =
00,
lim .lI-++co
& (x) =0 X
and ",<x) ;> tp(x), with equality holding for a sequence of values xi, x~, ... , x~, ... , x~ - co, corresponding to extreme points of the curve y = tp(x). In addition, ",<x) is everywhere twice differentiable, except at the points of contact.
Sec. 13]
EXTENSION OF THE CLASSICAL THEOREMS TO PROXIMATE ORDERS
41
At these points .of contact the first derivative y' = 'I?'(X) has angular points. By changing the function O(x) in an inessential manner these angles can be smoothed out so that the second derivative 'I?"(x) in each interval lies betweel). the upper and lower limits of the function 'I?"(x) at the corresponding points of contact. On the basis of inequality (1.57) we can assert that •
hm z-+ + 00
&" (x) 111 (
v
)
X
= O.
Now choose a majorant 'l?1(X) for the concave function 1f'1(X). Let tii.(x) be the smallest convex majorant for the function CPI(X) = cp(x) + 'l?1(X), and let 'l?z(x) be a majorant for tiiz(x). Then we have everywhere 'l?z{x) tii.(x) 'P(x) + "'l(X), while on some sequence of points x~, x~, ... , x~, ... tending to infinity and corresponding to extreme points of the curve y = Vi(x) we have the equality'l?2(x:J = tiir,(x~). On the other hand, since at the extreme points of the curve y = tii2(X) the equality tiiz(x) = cp(x) + 'l?1(X) holds, we have
>
>
&2(X~) = rp(x~)+ n1 (x~).
Thus we have constructed functions 'l?1(X) and 'l?z(x) satisfying conditions (a), (b), (b ' ) and such that, in addition,
rp (x)
-< 62(x) -
rp (x~) = &2 (x:) - &1 (x~)
&1 (x).
(x: -+ 00).
In the future L *(r) will denote a function having a representation In Le (r)
= 1)2 (In r) -
&1 (In r).
in which the functions "'1(X) and 'l?2(X) are concave functions satisfying conditions (a), (b), (b'). By p*(r) we mean the function defined by rpe(r)
that is, e( )
p r
=(i
= rPLe(r).
+ &2
(In r) - &, (In r) lnr •
The functions p*(r) clearly form a smaller class than the proximate orders p(r); however, as we have seen, this class is sufficient to construct a growth scale for the functions of finite order. More precisely, in the statement of Theorem 16 the proximate order can be replaced by p*(r). The function p*(r) will be called a strong proximate order. 13. ExteasioD of the Classical Theorems to Proximate Orders The classical theorems that were presented in the preceding sections on the CIOnnection between the growth of the function
In M,(r)
42
GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS
[CHAP. I
and the distribution of the zeros of the function fez) can be extended to the more precise description of the growth that is given by the proximate orders. The theorem relating the type of the function to the rate of decrease of the coefficients in the Taylor expansion can also be generalized. To formulate this theorem precisely, we introduce the function !p(/), defined to be the unique (for I > 10> solution of the equation t = rP (r). THEOREM
2'.
The type (I, of Ihe entire funclion 00
f (z) =
l: c"r' ,,-0
> 0)14& is git.V!n by the equation
with the proximate order p(r) (p
I
lim cp (n) "-+00 PROOF.
VI e,,1 = (o,?ef',
(1.06')
We first show that lim ,(kt) = k
, .. 00 ,(I)
Indeed, differentiating the equation In t
~
(1.58)
= p(r)1n r, with respect to In e, we obtain
:::: ==p; (r)r In r +p(r) and consequently .
dint
11m d-I ==p.
r-+oo
nr
This equation can also be written in the form lim d In , (t) _ '-+co dlnl -
.!.. p'
Thus we have the asymptotic inequality
(*-.)
d In t
< din cp (t) < (++ I) d In t.
Integrating from I to kt yields-the asymptotic inequality
(~ -.) In k < In ~ ~~)
aI'
In ICn I < arP (r) -
n In r.
Choosing r to be the root of the equation
n=
aprp(r)
we have
In Icn I or In (?
a, was arbitrary, (1.09')
We must show that equality holds in (1.09'). We define a by the equation n
lim cp (n) n~co
VIc:i =
1
(ape)p
and show that the assumption a < a, leads to a contradiction. Choose any number 0"1 between a and at (a < a1 < a,). We have for all sufficiently large n 1
IC I < { (alpe) P n
cp (n)
}n.
Using (1.58) this asymptotic inequality can be put into the form 1
Thus. for all n
> "0. eP
I cnr"" I < { (~) cp
alP
}IIrn
44
GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS
[CHAP. I
and the maximal term I-',(r) of the Maclaurin series of j(r) satisfies the inequality
(:) r" I
,.,0 and consequently
(1.61) I/(zO>I~M. The roots of the function w(z) are isolated points and, by the maximum principle inequality (1.61), must continue to hold at these points, that is, throughout the domain G. Note that equality cannot be attained at any interior point, unless j(z) is a constant. From this general principle we obtain a number of important tJ!.eorems that are frequently applied in various questions. THEOREM 20. Let f(z) be holomorphic in the domain G, whose boundary contains the point at infinity, and suppose that at all finite boundary points
I/(z) I 0, holomorphic inside the angle larg zl ~ 'TT/2p, and if on the sides of this angle THEOREM
then throughout the angle we have the inequality
!! (rei') I +k(6-6) 1J sin p (0 - 0t> 1
decreases monotonically as () -+ ()l limit (61) =
h:
+ O.
From this follows the existence of the
11m (pr (6, ( 1)],
....... +0
Similarly one shows the existence of the derivative from the left.
Sec. 16
55
ANALYTIC PROPERTIES OF THE INDICATOR FUNCTION
(c) The right hand derimtive is greater than or equal to the left hand derh'ath'e at each point:
This inequality is immediate if one writes (1.71) with 0 < 01 < Oa and passes to the limit as 0 -+- 01 - 0 and Oa -+- 01 + O. (d) The derivative h'+(O) is continuous from the right and the derivative h'-(O) is continuous from the left. PROOF.
Choose 01
< 0 < Oa.
It (0) - It (08) sin p (0 - 08 ) - h (83 ) :p. hsin(01) p (81 - 6s)
-
Just as we obtained (1.71) we can obtain
8- 6 2
.
h (!:IS> sin p - -1 sec p
61 - 88 8 - 88 sec p - 2 2
or It (8) - It (8a) ~ It (Ot> - h (fls) sin p (6 - 8S> ~ sin p (8 1 - 68 )
k(6 -01),
(1.73)
Passing to the limit as 0 -+- Oa - 0 we obtain h~
(as> ~ pr (6t •
6/1) -2pk( 03
-
( 1),
Similarly to (1.72) we obtain h'+(OI) < pr(OI' ( 3 ), It follows from the last two inequalities that
h~ (.6 3) - h~ (fjl) ~ - 2kp (I}s - ( 1);
(1.74)
and comparing this with property (c) we see that the functions
h~ (fJ)+ 2kpO and h~ (6)
+ 2kpO
are nondecreasing on the interval [01, 01 + q]. Thus the limits h'-(O - 0) and + 0) both exist. Passing to the limit as s -+- +0 in the equations
h~(O
f
8+.
h(8+s)-It(0)
s
=..!.. s
and
h~(Ij)dfj
+I h~ 8
h (8 -
~ ~ h (0)
=
(6) d6.
8-e
we obtain
"
h+ (6) = h+ (6
+ 0)
and h_ " (6) = h_ (6 - 0) .
(e) The function h(O) has a derivative at all points except possibly on a countable set.
S6
GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS
[CHAP. I
For the proof we write (1.74) in the form I
I
h+ (61) -
h_ (61) -
-< h_ (6.) I
2kp (6 s -
(1)
I
h_ (61) •
If 01 is a point of continuity for h'-(O), then I
h+ (a l )
I
h (6) h (a o) cos p (6 - 60>. Indeed, at a maximum point h'(Oo) = 0, while at a minimum h~(Oo) ;;> 0 ;;> (h'-0o). Putting 01 = 00 in (1.71) and passing to the limit as 0-+00 + 0, we obtain h (Os) - h (00)
sin p (Os - 00)
+ h (0)' Os - 0 0 sm p 2
0
sec p
Os -- On ........
2
-:?
.!.. h'
(" ) ........ 0 + Vo -:?
P
or for 0 < 03 - 00 < 'TTlp. Similarly one obtains an estimate to the left of the point 00 , Note that if 00 is a local maximum for h(O) and if h(Oo) = 0, then, by what has been shown, h(O) == 0 in some neighborhood of the point 00 , (g) For arbitrary IX and fJ (IX
< fJ)
(1.75)
•
and equality holds only for h (0) = a cos pO
+ b si n pO
(ex
-< 0 -< ~).
where a and h are constants. PROOF.
From inequality (1.71) it follows that as 0 -+ 01 :
(0 ) ~ h (Os) - h (°1) + 1 ...." sin p (Os ~ 0t>
2... h' P
+ h (0 ) tg p Os -2 °1 • 1
Interchanging 01 and 03 we have
2... h' p
-
(0» J
h(OIl)- h(Ot) -h(as)tgp 611 -;6 1 sin p (0, - 61)
and, after subtracting these inequalities,
1 I I 0l)]+[h(61) p[h_(Oa)-h+(
+ h(6).]tgp-2-> 6a -O 0. l
(1. 76)
Sec. 16]
57
ANALYTIC PROPERTIES OF THE INDICATOR FUNCTION
Subdivide the interval (rx, (J) by means of points 01 , O2 , ••• , 0n_l at which the derivative h'(O) exists; write inequality (1.76) for each of these subintervals and sum the results. We obtain _1 I
~
I
h_O)-h+(a)+p ~(h(6j)+h(6J+l)ltgp
8i + l
-
2
8 i
>0
j=O
=
60 ,
~
=
6n ). To obtain inequality (1.75) it is necessary to pass to the limit as max 18;0'-1 - 0;1- o. Suppose that for some rx and {J ({J > rx)
(a
sea, ~)=h~OO)-h~ (a)+pll
Since for rx
< 0 < (3
r II
J
h(/j)d6 =0.
II
I
I
sea, ~)=s(lJ, /j)+s(6,~)+h+(6)-h_(6)
and we have
I h_(&)+pll
f•h(&)d6=h+(a), I
II
or
+
h" (6) p2h (6) = O. Thus equality holds in (1.75) only for h (&) = a cos pO
+ IJ sir. p6.
Property (g) is characteristic for trigonometrically convex functions. In other words, the necessary and sufficient condition that the function h(O) be trigonometrically convex for some p on the interval (rx, (J) is that the function
f•
5(6) = h'(6)+p'l h(cp)dcp be nondecreasing on that interval. It is only necessary to prove the sufficiency. First assume that s(O) is a differentiable function. Then h" (&)
+ p'lh (6) =
(1.17)
5' (6).
(1.17') The Green's for the differential operator h" + on the interval 01 < 0 < 83 (0 < 01 < 03 < 17/ p), with the boundary conditions G(OI' 8) = G(03' 0) = 0, has the form sin p (811 - 8) sin p(.} - 6t ) for p 2h
function 18
0(4,6)=
11
I
4 < 6,
p sin p (81 _ 8s )
sin p (8 - 6t ) sin p (O~ p sin p (81 - 8s)
4)
For the Green's function see, for example, G. Sansone [1].
for '"
> 6.
58
GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS
[CHAP. I
The solution of equation (1.77) can be written in the form h (~) =
+
h (6 1) sin p (68 - 8) h (0 3) sin p (6 - 61) sin p (68 - 61) 8.
+ f O(~,
O)ds(~).
(I. 78)
I,
If s(O) is not a differentiable function then we approximate it by nondecreasing differentiable functions and pass to the limit in (1.78), and thus in this case too we obtain a representation of h(O) in the form (I.78). Since G(V', 8) < for 01 < 11', 0 < Os and since s(V') is a nondecreasing function, we have from (1.78)
°
(I. 79) h(O) is a trigonometrically convex function in every interval in which the function h'(O) + p2 h(O) dO is nondecreasing. Equality in (1.79) for even a single value of 0 7fI= 01, Os is only possible if the integral in (1.78) is equal to zero, that is, if s(O) is constant on the interval 01 < V' < 03 , or in other words, if h(O) = A cos pO + B sin pO. Formula (1.78) remains valid for 83 - 01 > 17/ p. It is only necessary that 03 - 01 should not be an integral multiple of 17/ p, since then the Green's function does not exist. We now show that every nondecreasing function S(O) determines a periodic, trigonometrically convex function h(O). For this we must construct the Green's function for the differential operator h W + p 2h with the periodic boundary conditions
r
0(0, fJ)
= 0 (21t,
,
fJ) and O 0 and for all sufficiently large values
'tr
n (t)
11'+1 (/-z)
For ar
n (/)dt tll+1(/_%)
< () < 277
'tr
,+1
f ~
t p- p - 1L (t)dt II-real
.".
Tr
(1.94) and therefore asymptotically rP+1
r
.
J "C'r
'If"
Ip-p- 1L
(t)dt
I 1 - re4' I
< 2rp+1L(r)
~
Ip-p-1dt = 2rpL(r)
I I - rea I
"
Thus, for 'YJ > 0 and 'YJ ~ the asymptotic inequality
(J ~
277 -
'YJ
AZP+1
f..,.
~,.
I f zp+l
n (I) dt tP + 1 (I _ relt)
Gf'
f ~P~:;:7 . "C'
II
and sufficiently small 15
t p- p - 1L (I) dt
t - rea
I
0,
we obtain
(1.95)
..!..8 rP(").
Gf'
Using (1.94) once more we obtain
..,.
I f Azp+l
u
~
tp-p- 1L (I) dt - tlei(p+ 1)9 {
t-re i6
r
•
up- p - 1 du } r PL (r) u-Id
I
II
(1.96)
66
GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS
Finally, for sufficiently small
(1
[CHAP. I
> 0 and sufficiently large 'T, 0:>
~
Ifup:::;u - fu;:~ dU/ < 8~' "
(1.97)
0
From inequalities (1.91), (1.93), (1.95), (1.96) and (1.97) it follows that asymptotically and uniformly in () for TJ ~ () .c;;;; 217 - 1]
fo ~:; dul < 0:>
/In V(re ifi )
-
tlrp(r)e i (p+l)I
er P(").
To complete the proof it only remains to compute this integral. 21 The reasoning we have used is not applicable in case p is an integer. In this case Lemmas 7 and 8 cannot be used to derive the asymptotic inequality (1.91). Also, (1.97) is not valid in this case since then the integral from zero to infinity does not exist. To study the case of integral p we prove the following lemma. LEMMA 9. Let the set {at} of points on the positive axis have density respect to the proximate order per) and let p be an integer (p ;;> I). Let
V,.(z)
a with
II a (~ ; p-1) ale>" IT a (:Ie ; p).
=
aJ:'
ak
e
P;
We then obtain
v (z) =
U~I) (z) U~2) (z).
and to complete the proof it is sufficient to quote Lemma 9. For p = p + I it is necessary to represent the functions u(1)(z) and in the form Ul 1) (z)= exp
(-fzp
U(II)(Z)
~ akP)u~t)(Z) a,,>r
and U(2)
(z) = exp
(f zp
~
ai P)
U~2) (z)
Gl>r
and then to use the same lemma. In conclusion, we construct a function that will play the fundamental role in the study of the generalized indicator function.
Sec. 17]
69
AUXILIARY FUNCTIONS
LEMMA 10. of the angle
Let p(r) be a given proximate order (p
> 0) and let the opening
61 0, then the generalized indicator satisfies the fundamental relation h (6 1) sin p (6 i - 6.) h (6a> sin p (6. - 61) +h(6 s) sinp(61 -6 g) 0 (1.69')
+
O).
By Lemma 10 there exists a function W(z} having the proximate order p(r), holomorphic and without zeros in the angle 01 " arg z " 03 , and such that the function In IW (rei'!) I r P''''
Sec. 18]
71
THE GENERALIZED INDICATOR
tends uniformly to the function HiO) on the interval 0 1 oe;; 0 oe;; 0 3 , Thus the func· tion f(z) W-l (z) tends to zero along the rays arg z = 01 (j = 1, 3) and by the Phragmen.Lindelof theorem it is bounded inside the angle 0 oe;; arg z oe;; 0 3 , It fol· lows from this that, for 0 1 oe;; 0 oe;; 0 3 and for arbitrary e > 0, h(O) oe;; He(O). It reo mains to pass to the limit as e -+ O. It follows from the fundamental relation that the generalized indicator h(B) satisfies properties (a)-(h) of § 16 which were derived from the fundamental relation. Now we shall derive some general properties ofthe growth offunctions along rays, where the function has the proximate order p(r). The results will, of course, be true also in the special case p(r) == p, that is, for functions of normal type with respect to the usual order p.22a THEOREM 28. /ff(z) is holomorphic and has the proximate order per) (p > 0) inside the angle « ('3 - '.) + h('3)('. R(E)
In 1I (re") 1
< (h(b)+a)rp(rl .
The following theorem on the connection between the type and the indicator follows at once from this theorem. THEOREM 29. The maximum value of the indicator h,(fJ) of the function f(z) on the interval at 0 if p < 1/2, and consequently we have. on a sequence r" too, the inequality 1011.(r,,)1 > 0 and also
Sec. 18]
72A
THE GENERALIZED INDICATOR
Hence in this case we obtain
-.- lnm(r) . InIJI(rn)1 Ii InIJI(rn)1 hl(O) hm ;> hm ;> m = -InM(r) 'ft- OO InIJI(-rn)1 'ft- OO hl(x)r:, (h (6) - e) rP (r) is valid. THEOREM 31 (V. BERNSTEIN [2]). Let the function f(z) be holomorphic and of order p(r) in the angle a " arg z " p. To arbitrary positive numbers e > 0 and 0 < '" < 1, and each fixed ray arg z = fJ, there correspond a number 8 > 0 and a sequence of intervals rn " r " rn(I + 8) (rn -+ 00), on each of which the inequality In If(re i9 ) I [h (6) -a) rP (r) is satisfied except perhaps on a set of measure not exceeding w~rft'
>
PROOF.
To each y
Without loss of generality we may asslime that fJ = 0 and h(O) = > 0 there is a sequence rn -+ 00 such that
In I f(r n) I> - yr"j-r.). Also, if we choose ~ > 0 sufficiently small, then for r > ry,'" depending only on ~ and y,
o.
IfJl < arc sin (2e~) and
In If(rei~) 1< (h (6)+ "() rP (r) < 2y,.,,n Z f(rn) •
Then CPn(O) = 1 and for Izl '" 2e8rn In I ~n (z) 1-< 3"( (r Iz I)p(rn + I -' I. It follows from Theorem 11 that the inequality
n+
In I ~n (z) I > - 3"(H
(=) (rn + 2eern)P
(rn + 2ewnl
is satisfied in Izl '" ~r n but outside exceptional circles the sum of whose radii is not greater than w~r n' Returning to the function f(z) we see that the asymptotic inequality
lnlj(r)l>
-y[I + 3H(-i-)(l +2eO)2p]r~(rfll
is satisfied on the whole interval (1 - 6)r. < r < (1 + 6)r., except perhaps for intervals the sum of whose lengths is less than 2w6r.. Using properties of the 22b There are many papers about Wiman's theorem. For a generalization of Wiman's theorem to meromo~c functions see GoI'dberg and Ostrovskii [1).
l2c The continuity of the indicator when p - 0 follows from a theorem of Griiin (see footnote at the end of 16 in Appendix VIII).
74
GENERAL THEORY OF THE GROWTIf OF ENTIRE FUN('''TIONS
function rP (1 +o)-2PrPlrl.
To complete the proof it is sufficient to choose y and b so small that
r[ 1 + 3H( ~)(1 + 2e8)2p] < e(l + 8)2p. 19. Plane Convex Sets
For p = I the indicator h«() has a simple geometric interpretation. To present it we shall need the basic properties of plane convex sets. A convex set is a nonempty closed set which contains the entire line segment joining each pair of points in the set. In particular, a line segment is a convex set. A single point is also considered to be a convex set. Clearly a nonempty intersection of convex sets is again convex. The intersection of all the convex sets containing a given set is thus the smallest convex set containing the given set. It is sometimes called the convex hull of the given set. It is easy to see that the convex hull is also equal to the intersection of all the half-planes containing the given set. We introduce the operation of arithmetic sum of convex sets. By the sum G1 + G2 of two convex sets is meant the set of all points of the form z = Zl + Z2' where Zl E G1 and Z2 E G2. Let us show that G1 + G2 is a convex set. Let z~ + z~ and z; + z; be two points of this sum. Since
+z;) = AZ~ +IlZ; + (A + II = 1, A>- 0, !l >- 0),
Z = A(~+z;)+!l(z:
AZ;+!lZ;
and since ).z~ + !-,z; E G1 and ).z~ + !-,z; E G'I. by the convexity of G1 and GI , we have Z E G1 + G'/.. It is clear that G1 + G2 is closed. We shall always assume in this section, without explicitly mentioning it each time, that our convex sets are bounded. An important concept is that of the supporting function of a convex set. By the supporting function of a convex set G we mean the function k(fJ)= sup (xcos6+ysin6)=supRe(ze-"). z+illEO
(1.102)
_EO
Since G is closed and bounded, this supremum is attained at some point of the convex set G. The line
x cos 6
+ y sin 6 -
(1.103)
k (6) = 0
will be denoted by 18, Clearly the line 18 has a point in common with G. Further, all points of G lie on one side of each such line, since from the definition of support function k(O) we have, for x + iy E G, x cos (j y sin (j - k (fJ) arg ZI' The line segment joining ZI and Z2 belongs to G and therefore lies on the same side of the supporting lines at the points ZI and Z2 as does the origin. It follows that the supporting line at Z2 has been rotated through a non-negative angle with respect to the supporting line through ZI. that is, O2 ;> 01 , Clearly at angular points of the boundary the function 0 = O(s) has jumps, while on line segments in the boundary O(s) is constant. The inverse function s(O) has jumps when the ray arg z = 0 is perpendicular to some line segment in the boundary and is continuous for all other values of O. The geometric interpretation of trigonometrically convex functions enables us to obtain the analytic properties of these functions that were established earlier. We shall obtain geometric interpretations of the quantities 8.
k~(6). k~(6) and s(61• 69)=k~(6:J-k~(61)+ fk(a)d6 8,
of which we spoke in § 16. Let us show that k(O) is a continuous function. Without loss of generality we can assume that the origin lies inside the convex set G. We denote by tS(~O) the length of the segment of the ray arg z = 0 + ~O between the supporting lines /" and 16+116 (Fig. 5). We then have k (0
+ AD) = k (6) sec (AO) + a(AO).
(1.113)
Sec. 19]
PLANE CONVEX SETS
79
To establish continuity it is sufficient to observe that 13(110) 1< d tg(M), where d is the diameter of the domain. We now investigate the differentiability of k(O) and we find the geometric significance of k'(O). Let p(O) denote the segment of the line Ie froni the point of support to ttoe foot of the perpendicular from the origin to the line Ie. It is easy to see that the radius-vector to a point of the curve r is a continuous function of sand consequently, at the extreme points, it is a continuous function of 0. It follows that p(O) is continuous at the extreme points. If the direction 0 is perpendicular to some segment on the boundary then we have
r.
p (0
+ 0) -
p (0 -
0) = d (0).
where d( 0) is the length of this segment. From simple geometric considerations (see Fig. 5) we have the inequality tg (M)(p(fJ+AfJ)-Asl (1. The smallest convex domain I, containing all the singularities of !p(z) is called the conjugate diagram of the function fez). It clearly lies inside the circle Izl O. It is easy to verify that this integral converges absolutely and uniformly in the domain (1.122) Reze- i8 h,(-6)+a.
>
The convergence of the integral follows at once from the asymptotic inequalities
Ie-de- i8 I< e -lla,(-81+e1 e and [ Ia
I!(tr ia ) I < e '
{-81+.!.]
e
2.
To prove (1.121) it is sufficient to show that the equation is valid in a part of the domain (1.122), for example, for Re Ze- il
> 30.
We show that for these values of z the series (1.118) can simply be substituted into (1. 121) and integrated termwise. Indeed, in that half-plane
Irife-iII < ra., . On the other hand, from the inequality
la,,1
M,{r)
1iI 0), and for every trigonometrically convex (with this p) function h(O) of period 21T. there exists an entire fUllction of proximate order p*(r) with indicator p*(O). An analogous statement is true for an arbitrary proximate order if p is a nonintegral number.
We note that the entire function with a given indicator which we shall construct in § 4 possesses a special regularity of growih. When the zeros of an entire function are regularly distributed, one can obtain an asymptotic expression not only for the function In Ij(re i8) I but also for the function argf(z), i.e .• one can obtain a representation for the entire function fez) itself. § 5 of the present chapter is devoted to this topic. In order that the expression argf(z) may have a definite meaning for the canonical product 00
f(z)
= IT a (~; k=l
a"
p)
(p = (p».
we define the function arg G(u;p) in the u plane, cut along the ray (1, +(0), to be zero on the upper side of the cut and then we extend in a continuous manner over the entire plane. The function arg G(u; p) that is defined uniquely in this way over the entire cut plane will have a jump equal to 21T at each point of the ray (\, +(0). We shaH say that a ray arg z = 0 is an "ordinary" ray if it satisfies the condition
. {-I· \1m 1m
h-+O ,.-+00
nCr. O-h. O+h)} () r P ,.
0•
All other rays will be called "exceptional" rays. The monotone nature of the function ~(O) implies that if the zeros of the functions are regularly distributed then the set of exceptional rays is at most denumerable. The basic result of § 5 can be formulated in the following way. THEOREM 4. Ijm is a regularly distributed set with index p(r) and iff(z, m) is its canonical function, then for any z belonging to an ordinary ray, and not belonging to some CO-set, the following asymptotic relation will hold:
In !(re i8 • !Jl) ~ J (6) r p(,,).
(2.08)
• This theorem is a generalization of a theorem of v. Bernstein [I; 31 on functions of normal type for p(r) - p.
Sec. I]
BASIC RESULTS
where
f•
J(6)=_K_ elp('-~-")dA(IjI) sin Kp I-a.. for a non integral p, and J(6)
= -I
f•e1p
(1-+) (6
-~)dA(IjI)+Aelp(8 -10>
9S
(2.09)
(2.10)
8-a..
for a p that is an integer. Here the convergence of r-p(r) Inf(re io ) to the limit J(O) will be uniform in 0 on any closed set IDl of ordinary rays.
We note that the multiplication of the function fez, 9l) by an exponential factor eP(z) in which the exponent P(z) is less than p will not invalidate the asymptotic equation (2.08). The exclusion of the CO-sets in Theorems I, 2, 4 and S is based on the Cartan estimate (see Chapter I, § 7). In this estimate the set of exceptional circles is not effectively constructed. It is for this reason that Theorems I, 2, 4 and S assert only that the asymptotic equations are valid outside of certain exceptional regions (CO-sets) but do not indicate how these regions can be constructed. However, with certain auxiliary limitations on the roots of the functions, one can give more precise information about the position of the circles of the exceptional set. We assume that the points can not come arbitrarily close to each other. More precisely, we assume that one of the following conditions (C) or (C') holds. (C) There exists a number d > 0 such that the circles of radii p (I 11ft I)
Tft=dlanl
1---II
with centers at the points an do not intersect. (C') The points an lie inside angles with a common vertex at the origin but with no other points in common, which are such that if one arranges the points of the set {an} within anyone of these angles in the order of increasing moduli, then for all points which lie inside the same angle it is true that
Iak+1I-1 akl > d Iakl l - P ( 111,,1) for some! d > O. In the conditions (C) and (C') the proximate order p(r) has the same meaning as in condition (A). A regular point set {an} satisfying one of the conditions (C) or (C') will be called a regular, or more briefty an R-set, while the circles Iz - ani 0 one obtains the limit i8
. InTI(re)· = -7t~ \ 1m - e ipl9-,!,-1t\ r ~ co r P (T) sin 7tp T
where the convergence is uniform relative to () in (tp Equating the real parts, we find that
+ rt
~ () ~ tp
+ 27T -
'YJ)'
'9
. In I TI (reI) I = -7t~ \ 1m - - COS r, ('J"- ' I'" .- _.). . T~OO r P IT) sin 7tp , ,
The case next in complexity occurs when the zeros of the canonical product II(z) are located on a finite number of rays arg z ='1'; (j = 0, I, ... ,m), and the density of the set of zeros exists on each ray, that is, when we have the limits Ttj (r) ~ r~co r P .
Aj = 11m
(j=O, I, ...• m).
where nir) is the number of zeros of the product II(z) in the interval (0. r) on the ray arg z = tp;. In this case the function II(z) is the product of functions II ;(z) whose roots lie on a single ray arg z = tp;. This case can be reduced to the preceding one. and we have the limits lim
T~OO
In I II (re ffl ) I r P(r)
7t:' = - - ~. A· coso (5 -w.-1t) sin 7tp ~ J ' 'J J~O
where the convergence is uniform relative to () if I() - tp;1 ~ rt > 0 (j = O. I, ...• m). If one writes the sum of the right hand side of this equation by means of a Stieltjes integral, one obtains formula (2.04) for this particular case of regular distribution of zeros. The transition to the general case is based on the following lemma. LEMMA
I.
Let us assume that the set {an} of the zeros of the canonical
product co
II(z) =
IT a (:n; p) 1
has a density with index per), i.e., there exists the limit A _
a -
I'
rt (r) Im-C-)' rPr
and suppose that p = limr-<x; per) is not an integer. Let us denote by II"(z) another canonical product 00
IP (z) =
IT a ( an~ ; p) , 1
in which la~1
= la,,1 and larg a~
- arg ani
< «5.
Then for every
E
> 0 and rt > 0
Sec. 2]
ENTIRE FUNCTIONS OF NONINTEGRAL ORDER
there exists a ()
>
99
°
such that
lin I II(z) I-In I n~(z) II < er W ) for al/ z that do not belong to some exceptional set of circles C with upper linear density less than?J. (Here b depends only on the numbers E and ?J, while the exceptional set of circles C depends on the sets {an} and {a~}.) This lemma makes it possible to replace, with an arbitrary degree of asymptotic precision, a given canonical product by another one the roots of which lie on a finite number of rays. The proof of this lemma is somewhat cumbersome and we shall give it later in order not to interrupt the train of thought. Let us now continue the interrupted proof by making use of the lemma. Thus let us assume that fez) is an entire function of integral order p, and that the set of its zeros has an angular density !:::.,6 with index per). Let the quantity H(fJ) be determined by formula (2.04). We construct the sum
Sm'(~) =
5l:
m-l
~ cos P (6 -lJIj -
1tp
j=O
r.) [A OJ+1) - A (~J)]
(Ym = Yo + 2r.).
It is obvious that for every given positive number E there exists a positive number b such that if max; IVim - Vi;1 < b (j = 0, 1,2, ... , m - I), then (0
-< /j < 27t).
(2.13)
We assume, as has been mentioned above, that the function f(z) is a canonical product co
f(z)
= II a (a:;
p)
(p-I
r., u (j= 0, I, ... , m). Because the sets 10 - 'I'il u and 10 - 'I'il u cover the entire interval [0, 217], we see that the inequality (2.16) is satisfied when re iO E C = C + C', where r(C) < 'YJ. One can thus construct an exceptional set C of circles, with an arbitrarily small upper linear density, outside of which the inequality (2.16) will be satisfied for any preassigned positive E. Making use of this fact, we shall construct a set of circles, of zero linear density, outside of which set the required asymptotic equation will hold. For this purpose, we select two arbitrary null sequences of positive numbers {e,l} and {'YJ,I} and construct a sequence of sets C ,I (p = I, 2, ...) of circles such that p*(C,,) < 'YJ,I and that, for reie E C ,I,
>
>
>
>
I'-P "'lin It(re i9 ) 1- H (IJ) i
< Bp.
(2.17)
We shall denote by l:r rf the sum of the radii of those circles of the set C,I whose centers lie in the circle Izi < r. Since C ,,) < 'YJ", it follows that for r sufficiently large we have the inequality
r(
~r~
< 2"1,1".
(2.18)
Sec. 2]
ENTIRE FUNCTIONS OF NONINTEGRAL ORDER
101
Let us select a sequence of positive numbers R., (p = 1,2, ... ) such that R1J+l > (p + I)R" and that the inequality (2.18) is satisfied for r > R". From each set C" we pick out those circles whose centers lie in the ring
Rp" Izl < Rp+1' and we denote the set of these circles by C. This set is a CO-set. Indeed, if we reorder the circles of C and denote by r; the radius of the jth circle, we find that for we have the inequality
.!.r~ ~r rj < 2.;"11... p +3. :"1:1... p + ... + 2'11p < 2"1tp e + 21Jp • Therefore,
p*(C) = O.
Furthermore, if reiO E C, and if r >~, the inequality (2.17) is satisfied. In other words, if Z E C, then the quantity
hr, r(fJ) = rP (f') In I/(re i ') I converges uniformly to H«(). This completes the proof of the theorem.' We shall now prove Lemma 1. I. Let the positive numbers (j, T and {J be given such that (j < I, T > 1, and {J does not exceed either I - (j or 7' - 1. We shall represent the infinite product co
IT a (a:; p)
I(z) =
1
as a product of the following functions:
1o (z)=
IT a (a:; p), lo,~(z)= II
II a (a:; p),
./(z)=
IlInlu-
a (aZn ;
p),
• It is useful to note still another formula for the function H(8),
f
2K
H(6)
= 5-I~ nap
cos P (16-
"'1-K)d~ I we have the asymptotic relations lin
Ila (z)11 < rorP(r).
lin 1,/(z)l/
< rorP(r),
lin
II: (z) II < rorP(r),
lin I,l(z) I I
< iorP(r)
and hence,
I In Ila(z)I-ln I/!(z) II < i- rP (r), Ilnl./(z)I-lnIJ5(z)11
< irP(r).
(2.20)
3. To estimate the third and fourth term we shall make use of the uniform continuity of the function
in sand
In I 0 (se- i9 ; p) I
°
(s>O)
within the regions
(1 -
p)-1
-< s -< a-I,
.. -1
-< -< (1 + ~)-I. S
From the existence of the density 6. of the set of zeros {an} (a" ~ 0) it follows that for some c > 6. n (r)
< crP
If we select 15 > 0 so small that for (1 - {f)-I and for IOl - 02 1 < 15 the inequality
(2.21)
(r) •
< s < 0--1, or ,,-1 .;;
lin 10 (se i81 ; p) I-In I 0 (se iS.; p) II
< l~~P
S
nJ (21n ~ -In 2-1).
(2.25)
Since each exceptional circle contains a point a .. from the ring K i • the centers of these circles lie either in KI or in one of the adjacent rings KI-l and Kl+l' Let us denote by C the set of aU exceptional circles that correspond to the various rings XI (j = 1,2•...) and by T" (k = 1,2, ...) the radii of these circles. We now shall give an estimate of the upper linear density of this set. Let R be an arbitrary number greater than 1 - p, and let m be such R.. < R < R..+1' Then
~R'k
",+2
-< J=O ~ 2~'!RJ < ~ (I +~) Rm+2 -< ~ (I + ~)S(I -
~)-2 R.
On the left side of this inequality stands the sum of the radii of those circles of C whose centers lie in the circle Izl ..;;;; R. Therefore,
p·(C)
-< ~ (1 + ~)S(l- ~)-2.
Sec. 2]
105
ENTIRE FUNCTIONS OF NONINTEGRAL ORDER
The inequality (2.25) yields a lower bound of the function I'f'RjZ)1 inside the circle Ki but outside the set C which we have constructed. Let us now proceed to estimate the function I'PR,iZ) I outside of this set for an arbitrary R. For this purpose we express the function 'PR.P(Z), for R.,. (n m +nm+1) [21n ~ -
1 -In 10).
Combining this result with (2.24) we see that lin I YR. ~ (z)
+
II < (n", + nm+1) [2 In + 4] .
(2.26)
Let us now find bounds for the function
4>R.~(Z)=
II
(l-~)RR. ~ (z)1 = In I'~R. ~ (z)1
+Re{
~
(~+ ~: + '" + ZPp)}.
(1-Ii)B 0, for a sufficiently small (J > 0, and for a sufficiently large T ;> I, we have the asymptotic relations
lIn II,., .. (z) II < lIn I./r (z) II
0).
> 0 sufficiently small, \In I ~r.~(z) I-In 14~.~ (z)11 < ~ rP(r).
(2.35)
From the inequalities (2.31), (2.32), (2.33), (2.34) and (2.35) it follows that for a sufficiently small positive 15 and for z outside of the set C we have the inequality lin I Ir(z) I-In II~(z)
II
0, 'fJ > 0,
rP(")
and for a
(2.37)
will be satisfied by all values of z which do not belong to an exceptional set C of circles with upper linear density less than TJ/2. For the function F,(z) it was established that if (1 > 0 and larg z - 1p;1 ;> (1 (j = 0, I, ... ,n), one can find a number r •. a such that for r ;> r •. a
I ,-p (,,) In / I; (re ie) 1- Sn (6) I < ; .
(2.38)
From (2.36), (2.37) and (2.38) it follows that if z E C, and if larg z - 'P;I ;>
(1
(j = 0, 1,2, ... , II), then
I r-' (,,) In /1" (re ll) I -Hl (6) I < a.
(2.39)
Sec. 3]
113
ENTIRE FUNCTIONS OF INTEGRAL ORDER
After making a second subdivision of the interval (0, 211" + 15) by means of the points 'Pi (j = 0, I, 2, ... ,17) in such a manner that 'P~ = 'P~ + 211", 1'I}i + I - 11'1/ < 15, and after choosing a positive a sufficiently small, we get the result that the sets 10 - 'Pil ;> a and 10 - 'Pi I > a (j = 0, I, ... , n) cover the entire interval [0, 211"). Making use of this second subdivision, we find that the inequality (2.39) is satisfied if z = reiO does not lie in a certain set C' of circles whose upper density is less than "1/2 and if 10 - 11';1 ;> a (j = 0, I, ... ,n). In this manner the inequality (2.39) will hold everywhere except for a set of circles C = C + C', where
p.(C) < '1\. In exactly the same way as in § 2 for the proof of Theorem 1 (pages 100-101), one next constructs the CO-set outside of which the relation (2.40) holds uniformly relative to O. We note once more that in this derivation we have made use only of the existence of the angular density of the set of roots. Lemma 5 has thus been proved. Let us now proceed to the proof of Theorem 2. By hypothesis, the point set {a,,} satisfies the condition (B) (see page 91). If one makes use of this condition, the representation (2.29) and the equations (2.30) and (2.40) yield at once for z E CO the uniform convergence of the limit
11m In I/(r~) I r
-+ 00
,.,
(r)
= H (6),
(2.05)
where
H(6)
= 't,cosp(6- 6,)
- f•
[(6- +-1t)sinp(6-+>++coSP(fJ-+>JdA(+>.
I-a.
Let us transform this expression to the form (2.06). For this purpose we note that for an integer
• f [! cosp(fJ -+>+1t sinp(6- +>] dA(.~)
I-I..
a.
=
JI! COSP(6-+)+1tsinp(6-+)]dA('~) = AcospfJ+B sinp8.
114
ENTIRE FUNCTIONS WITH ANGULAR DENSITY
[CHAP. II
where A and B are constants, and we find that 8
f (6 -I~)sln p(6 -~)
H (6) = Al cosp6 +81 sin p6 +
dll.(,?).
8-llr
Indeed, the ratio of the diameter do of an arbitrary region n, consisting of exceptional circles of the CO-set, to the distance of this region from the origin tends to zero when this distance is increased. From the maximum principle it follows that the value of the function In 1/(z)1 at any point of such an exceptional region does not exceed its maximum on the boundary of this region. But on the boundary of the exceptional region the asymptotic formula for h, .•(O) is valid, and, taking into consideration the continuity of the function H(O), we find that for re i8 E n the following asymptotic inequality applies: In If (re 48 )
1< (H(6) + ; )(r+dg)p(r+dg ) < (H(6)+e)rp(r).
On each ray arg:: = 0 there are points arbitrarily far away from the origin which are not covered by the circles of the exceptional CO-set, and hence for every 0 (18) lim Inll(re )1 =H(6). r-+oo r P (r) 7
The equation (2.41) can be obtained directly from the condition (B).
Sec. 3]
115
ENTIRE FUNCTIONS OF INTEGRAL ORDER
It is interesting to note that the regions n consisting of exceptional circles and not containing zeros of the function j~z) can be omitted from the exceptional set. Indeed, applying the minimum principle to the function In If T n"(r,,, 1\')-n,,(r"_I' ,~)< 1L\(I\')+.)(rP(rk) - rP(rk-l' " It-I From this and from (2.47) we obtain
+2dll ).
.. +1
and, in view of (2.46), -\.-n(r. IjI) /' A(,)+ 1m ..lItr) ~ '-A." ••
r .. co r·
118
ENTIRE FUNCTIONS WITH ANGULAR DENSITY
[CHAP. II
Analogously we obtain . . . . 04(') . n (r, <jI) ..,;?~ I1m 't -E,
~
and, since
E
r Plr )
is arbitrary, we see that
lim n (r, <jI) r -+ 00 rP Ir)
= tl (1jI).
Thus we have constructed a set with the given angular density and have proved Theorem 3 for the case of non integral order. We call attention to the fact that the functionj(z) which we have constructed not only possesses the given indicator h(O) but also satisfies the asymptotic equation
In If(re i &) I ~h(6)r plr) in the complement of some CO-set. 3. The case of integral order is considerably more complicated. In this case it is not sufficient to construct a point set {an} with the given angular density
tl(,~)=
2!P
f
[h'('~+O)-h'(+O)+p2 '"
h(IP) dIP]. (2.48) o It is necessary to construct the set so that the condition (B) is satisfied. Furthermore, comparing the expression (1.82) for the given function h(O) with the expression (2.06) of the indicator, we see that for some cp the limit
'Cr8 -ip9f
= r-+oo lim L (1 ) {cp +.!. r P
~ ai' }
(2.49)
la" I (Z)
=
lJ-~
"
120
ENTIRE FUNCTIONS WITH ANGULAR DENSITY
[CHAP. II
If p(r) = const, then one may choose for 91' the empty set, and for the function 9'(z) the exponential function exp c,z' (C, = T~-ip8I). In the general case, for an arbitrary p*(r) the function 9'(z) must (because of Theorem 2) satisfy in the complement of some CO-set the asymptotic equation
lnl9'(rei')1 ~ "",COS p( (J - (J,)r>. Hence, the case when p is an integer has been reduced to the construction of a set 91' with the indicated properties, or, what is the same thing, of a function !p{z) which is the analog (for an arbitrary proximate order p*(r» of the function exp c,z'. For the construction of the desired function, we note that from the definition of a strong proximate order p*(r) and of the corresponding function L *(r) it follows that r P+ 1 [L·(r»), =r P [1}~(lnr)-O~(lnr)]
,a_(II1",-&, 0 and 0 < t < E. Integration by parts yields r
_1_ pL*(r)
S
dn(t)
tP
=
r
nCr)
pr P(r)
r
+_1_ {SIL*(t»)'dt+ S ,(t) dt}, L *(r) tp+l
o
• I (2.54) 1 where !pet) = net) - t P + [L *(1»)' = 0(1). Let us include in the set of the points bl'; the point b defined by the equation co
lrP = L*(a)-
S:p~!dt . •
Having constructed 15(r) for the corresponding canonical product, we obtain l5(r) = I + 0(1). By multiplying each of the numbers bltl and b by Tjl/p ei6r, we obtain a set {bit} of zero density such that
· {I I1m L*(r) r-+co
P
~
~ Iblcl the following asymptotic relation will hold uniformly:
°
n
arg l(re i8 ) ~ -J_1t- ~,:1, sin p (6 n7tp ~
S
J
~'i -It).
(2.60)
i~O
In this case of regular distribution of the zeros we choose for a given positive a positive b so small that, for I'PHI - 'P;I < b (j = 0, I, ... , n) and 'Pn+1 = 'Po + 217, the sum on the right hand side of this equation will differ from the corresponding Stieltjes integral by less than E/2. Furthermore, if b is so small that for () E 9)1 E
I arg 1(re4~) -
arg f (re la) I
< ; rp(r).
then for () E 9)1 and () not belonging to the interval I'P; - ()I for some r.,,9R and r > r.,,911 the inequality
I
arg I(rei~) -
f
8
1trP(r) Sin 1tp
sin p (6 -.} -It) d~ ('f)
< 'YJ we shall have
I
-",(or)ln(1+8)r.
If"
f 0
n't(t) dt
126
ENTIRE FUNCTIONS WITH ANGULAR DENSITY
[CHAP. II
From these inequalities it follows that 1)
(ar
nz(t)
IlnIF.,(z)1I < In ( 1 +"6 nz (6r) + Jo -t- dt. LEMMA 8. If the poillt set {an} is an R-sel and if F~(z) is the same as in Lemma 7, thell for erer), pusitire E, for a sufficiently small posit ire number ~, and for allY Z I)'ill~ outside the exceptional C Il"circles and outside some circle 1=1 ~ r., we hare the inequality
lIn I Fa (z) II < ar P (r).
(2.62)
PROOF. Suppose the condition (C) (see § I) is satisfied, and suppose that the point;; does not belong to any of the circles mentioned in that condition. With =as center let us draw a circle K, of arbitrary radius I > O. Since the center of this circle does not lie in any of the exceptional circles. the radii of the exceptional circles with centers inside this circle are less than t. Since no two of the exceptional circles have a common point, the number n.(I) of their centers within the circle K, is less than the ratio of the area of a circle of radius 2t to the area of the smallest one of the exceptional circles with centers in XI' If we denote this smallest radius by I"., then for t < r we have
Lt "
> d(r -
plt+r)
t)(r +t)- - 2 -
and consequently, for t '" ~r and for sufficiently large r, 47ttl!(r+t)P(r H )
n,,(t)
dk be satisfied. Then n.{t) < (Ild)t and if Iz - a ..1 > d. (d. > d, n = 1,2, ... ) the inequality (2.62) will hold.
Sec. 6]
ENTIRE FUNCTIONS WITH REGULAR SETS OF ZEROS
127
We need to show that the second inequality is valid everywhere outside the CR-circles when r > r;. For this purpose we pick a z outside the CR-circles such that Izl > 2r;, and we define the function Fa(z,u)=
II (l_zt
I an - 81 " ~r
U ).
n
It is obvious that F~(z, 0) = F~(z), and that the inequality In IFiz, u)1 < 0 will hold if ~ < t and lui < ~r. This implies that within this circle but outside the exceptional CO-set the function
<Pa (z, u) =
/(z+u,9l) Fa (z, u)
satisfies the inequality In I<Pa (z, u) I
> (H~ ('J.) -
e) Iz+ u IP (1-+"1),
in which 1p = arg (z + u). From the continuity of Hm(1p) it follows that for a sufficiently small positive ~ we have In I<Pa (z. u) I > (H!R (6) -
(2.63)
2e] rP (r)
in the same region. We shall now show that for sufficiently large values of r this inequality is valid when u = 0 even if z belongs to the CO-set. Indeed, for a given ~ > 0, and a sufficiently large r, the sum of the diameters of the circles of the CO-set which lie in the circle lui < ~r is less than ~r, and hence there exists a circle lui = 7' (0 < 7' < ~r) that does not intersect the CO-set. The inequality (2.63) is satisfied on the circumference lui = 7', but since the function rp~(z, u) has no zeros in the circle lui 0 such that if 102 - 011 < lJ. , > '" and ,ei81 , rife, do not belong to some CO-set (or if r does not belong to an EO-set that is independent of e). then
I h"
r
(6 1)
-
h"
r
(6 2)
I < I.
For r 0 and 101 - 021 < lJ2 the same inequality will hold. It is sufficient to take 6 = min (61, lJz). Thus if the set {a/c} of the
Sec. 7]
129
THEOREMS OF EQUICONTINUITY
zeros of an entire function of proximate order p(r) has a density. the assertion of Theorem 6 follows from Lemma I, as has just been shown. 13 In the proof of Lemma I we made use of the density of the set of zeros of the function only in the estimate of the quantity lIn IR,Il(z)lI. For the proof of Theorem 6 without auxiliary hypotheses on the set of zeros, it is therefore natural to make use of the general estimate for lIn "1 R,II(Z) 1 given in Lemma 2. This estimate involves the quantities n", and n",H' namely. the number of zeros of j(z) in certain rings Km: Rm{l - (J) < Izl 0 and e > 0 be given. We choose {3 so small that
2PHC'3~ 0 there exists a tJ > 0 such that if lOt - 02 1 < 15, then (2.74) will hold. For the construction of such a set E we choose for the given "I two sequences of positive numbers: £1' £2' • . . • converging to zero, and "It. "12' .•• such that 00
To each pair of numbers CIe , "lie there corresponds a set Ek with upper relative measure less than "lie' and a positive number tJ(t) such that if r E Ele and 181 - 02 1 < 15(1), then lIn I f(re iO .) I-In I f(re i9.) II
< skrP
(rl.
(2.75)
We improve the sets E" somewhat by dropping from each Ele the part belonging to a certain interval (0, rle)' and we do this in such a way that the remaining set E" satisfies the condition mes (E: I
< 2'1"",r
(r
> 0).
Sec. 7]
133
THEOREMS OF EQUICONTINUITY
Then it is obvious that the upper relative measure of the set
E;
will not exceed 'YJ. The set and the number bill are selected in such a way that if, > 'k' r E E. and 102 - 011 < b~1) the inequality (2.75) will hold. On the other hand. if Izl < rIc. and 1=1 E £1' the function In 1/(z)1 is uniformly continuous. and hence the inequality (2.75) will be satisfied for some ,}~2). 19) - 921< 2>, r " rk , and r E E. If we choose 6k = min { BP>, BP>}, we find that (2.75) is satisfied if r E E and 191 - 921< BIe... This implies the equicontinuityof the family of functions "",(9) in 9 if r E E. The theorem has thus been established. Let us now proceed with the proof of the more general Theorem 7 on the equicontinuity of the family h F.r(O) of functions which are holomorphic inside some angle 0( ~ arg z ~ p. We shall first prove three lemmas.
61
LEM~A
9.
Let $(z) be a holomorphic function in the circle
Izl
~
R, and let
max I 0 the number n(r, ex in the sector ex + ~ < arg z < {J inequality
+ ~, {J ~,
~)
of the zeros of this function
Izl .s;; r, will satisfy the asymptotic
in which h=
. max
1h,l 0 the family of functions h
P.,. (6) = In IF(re rP,r)
il ) I
is equicontinuous in 8 if 181 " 0'3 < TT/2, and r E E'I' where E'I is some set of positive numbers with an upper relative measure less than TJ. This completes the proof of the theorem.
CHAPTER
III
FUNCTIONS OF COMPLETELY REGULAR GROWTH In this chapter we shall investigate a problem which can be considered as the inverse problem to that of Chapter II. There we established that regularity of the distribution of the zeros of an entire function implies regularity of growth. In this chapter we shall show that regularity of growth implies regularity of the distribution of the zeros of the function. l A function F(z) that is holomorphic and of proximate order per) within some angle (IJ l , IJ 2 ) will be called a function of completely regular growth on the ray arg z = IJ if the limit i8
hF (6) = lim In IF (re ) 1 r~oo
rP
(r)
exists under the condition that r goes to infinity by taking on all positive values except possibly those of a set of zero relative measure (an EO-set). We shall sometimes write lim· to express this situation. We shall say that a function F(z) is of completely regular growth on some set of rays R9Jl orn is the set of values of IJ) if the function h
'8
F. r
(6) = In I F(re t )1 rP(r)
converges uniformly to hF(f) for f) E WI when r goes to infinity by taking on all positive values except possibly for a set E9Jl of zero relative measure, this set being the same for all rays R9Jl. The set E9Jl will be called the exceptional set for the given function. We shall say that F(z) is a function of completely regular growth within the angle (0 1, ( 2) if this is true for every closed interior angle, and we say simply that F(z) is of completely regular growth if it is an entire function and is of regular growth in the entire plane, i.e., for 0 - 00, In IF (re iS ) I ~ h,1 (6) r P (r) •
We shall also prove that F(z) is a function of completely regular growth on every ray which is a limiting ray of the set of rays along which F(z) is of completely regular growth. THEOREM I. If a holomorphic function of proximate order pCr) within an angle (IX. (J) is a functil'n of complete~v regular growth on each ray of some set R<m belonging to the illlerior angle (IX < 0l 0 arbitrarily. Let us assume that we have chosen the numbers '1' '2' ... , rn-1' Then we pick, n so that' ;> 'fI' and the following inequality is satisfied mes(Ek'"-I)
< ~1J"r,
k
< 1J"r
(Er denotes, as before, the intersection of the set E with the interval (0,
(3.03)
r».
142
FUNCTIONS OF COMPLETELY REGULAR GROWTH
[CHAP. III
Such a choice is obviously possible. After the sequence {r n} has been selected we determine a set E such that for r n < r < r n+l' E = En. From (3.03) it follows that for rn < r < rn+ I the inequality mes E' < (2"'n + "'n_l)r will hold and hence that m*(E) = O. Furthermore, it follows from (3.02) that for r > r nand r E E, we have the inequality I hF. r (6) - hF(6) I En (6 E Wt). Thus F(z) is of completely regular growth on WI. In subsection 3 we proved a statement which we shall use below. For this reason we will formulate it as a separate lemma.
0 and 1J > 0 there exists a set E •. ~, whose upper relatire measure is iess than Yj, such that
0 and 101 < 2d we have the inequality Ih~O)1 < £/4 and that, in view of the asymptotic inequality In IF (re") I
< (h p (6) + ~) ,p
(r) ,
we have the following asymptotic inequality within the angle
InlF(reil)l
(3.11)
,
Let a" be the zeros of F(z). We set cpp (z)
=
II (1 - : )
a Ell n
n
p
and
9p (z) = F (z) 4!;1 (z). From (3.11) it follows that if P is greater than some fixed number Po, then
Inl4!p(z)1
(t)1 t
dt
o'.
+
J r
Inl W(t)1 dt ~ p-1hF(0).
lim r-P (r) r~oo
0
t
By a rotation of the plane one can superpose any ray arg z = () of the angle f3 onto the positive ray, and thus obtain the inequality
oc ~ () ~
r
. r hmr-p(r) r--+oo
In I F (tei~)1
t
dt>p-1hF(fJ)·
0-
It is sufficient to combine this inequality with the inequality (3.06) to complete the proof of the lemma. We note the following consequence of the lemma. COROLLARY. If F(=) is a holomorphic function of proximate order p(r) within the angle oc ~ arg z ~ {3 and of completely regular growth on the ray arg z = e, IX < e < (3, then the follo ..... ing limit exists:
f J}(fJ)'!!.=p-2hF(~)· r
lim r-+oo
r-~(r)
r,
t
(3.05')
A GENERALIZED FORMULA OF JENSEN
Sec. 2]
149
The truth of this assertion follows directly from Lemma 2 and property (5) of the definition of proximate order (Chapter I, page 35). Lemma 2 and its corollary have valid converses, in which one does not have to assume that F(z) is holomorphic.
Suppose that the function F(t) is defined for t
LEMMA 3.
h p (0) =
Furthermore,
if lim
~
0 and that
lim In I F (r) I •
T~OO
rP\T)
r-p("'J~ = p-lhp (0)
(3.16)
,.~oo
or
r
lim
"~.JO
r Jr dtt =
r-P (T)
• o
P
o-'!h (0)
(3.16')
IF'
then the limit . "In IF(r)1 ---''-;-'--'-
I 1m' T~OO
rP (r)
exists. PROOF. The condition (3.16') is a consequence of (3.16). Hence it is sufficient to give the proof under the assumption that condition (3.16') holds. Without loss of generality we shall assume in the sequel that hF(O) = O. Let y be some positive number and E be the set of positive numbers on which
rP
IT)
(3.25)
where 8
sp(~, 6)=(h~('J)-h~(&)+p\1
f hF(ql)d~l
(3.26)
&
The exceptional denumerable set can only consist of points for which h;,(O + 0) rF h;' 0 and k < 0, we obtain in an analogous way the inequality NCr,
a, 6) > _1_ S (n 2'1tp3
r P(f')
F
,
IJ)-I.
From these two inequalities it follows that
r
N (r, &, 6)
1m
,'-+ co
rP (f',
1
n
(3.33)
= . I, the fact that the function nCr, .{}, 0) is monotone implies the inequality
f
k··
n (r,3, 6) In k ~ __1_ r P ,rl
-..::::. rP ITI
n (t, 3, 8) dt
t
r
=
N (kr, 3, 6) r P lrl
N (r, -
a, 6)
rP (r)
,
and in view of (3.34) we obtain - . n (r, 3, 6) ./ kP -
11m r-+co
rP(r)
Taking the limit a.;:" -
1
1
~----SF
In k
27t p3
(n"', f'). J
I, we have
f;-
1m r-+co
n (r, 3, ~, ./ _1_ r
Plr)
:::".
2
'lTP
Sf'
(1\ I) 11 ,
J •
n,
6)- N(kr, D, 0),
Analogously, making use of the inequality r
nCr, n, O)lo!
~
f
n(t'tf), 6) dt=N(r,
kT
which holds for 0
< k < I, we find that . n (r, 3, 8)........ _1_ (1\ 6) \ 1m ) -? SF v, • r Plr 2'ITp
r -+ co
Thus the limit (3.25) exists for those values of {} and 0 for which the derivatives h'({}) and h'(O) exist, i.e., for all except possibly a denumerable set. This proves the theorem. COROLLARY. In order that the set oJzeros oj aJunction oj completely regular growth may have a density within some angle (at, fJ), it is necessary and sufficient that its indicator be sinuSoidal within this a,.gle.
Indeed, if the indicator is sinusoidal, then Seat, fJ) == 0, and in view of (3.25) the density of the set of zeros is equal to zero. Conversely, a zero density of the set of zeros implies that s( at, fJ) == 0, and by property (g) of the indicator (Chapter I, page 56), it will be sinusoidal. From the theorem just established, and from Theorem I, Chapter II, it follows that for nonintegral p, the class of entire functions with regularly distributed zeros and the class of functions of regular growth coincide.
156
FUNCTIONS OF COMPLETELY REGULAR GROWTH
[CHAP. III
We shall prove that this is also the case for entire functions /(z) of integral order. By Theorem 3 the set of zeros of the function/(z) has a density. But in this case, in line with the definition of a regularly distributed set, one must prove also that the foUowin~ limit exists:
}~"!., L ~f) {a +
+~
a;p},
lanl 0, except possibly for a set EO of zero relative measure, we have the inequality In I ~ (reiD) I> [h, (I}) -. e) rPI"). Thus for all values of r belonging to th~ set E' - EO. E' of positive upper relative measure we have In I~ (reiD) / (reiD) I [h f (6)+ h~ (&) - 2eJ rPlr). (3.38)
>
Combining (3.37) and (3.38) we obtain the following theorem. THEOREM 5. If the functions f(z) and rp(z) are holomorphic within the angle (01' (2) and if one of them is of completely regular growth, then the indicator of the product of these functions is equal to the sum of their indicators.4a 4a Azarin [1] has proved the following converse theorem. If .f(z) is hohJmorphic and har proximate order p(r) iMide an angle (II •• llUo and if for every hoIomorphic jrmction .,(z) of proximate order p(r) the indicator of the product j{z).,(z) equals the .rum of the indicaton, the,. j{z) is of completely reguku growth.
FUNCTIONS OF COMPLETELY REGULAR GROWTH
160
[CHAP. W
Let US also note that if both factors are of completely regular growth on some ray, then the product is of completely regular growth on that ray. If the entire function feZ) satisfies the condition •
11m
In M, (r) rP
1'-+00
t,.)
=0.
then it follows from Lemma 4, Chapter I, that .
n,(r)
11m ---;trI = 0,
1'-+00
r
and hence the set of zeros of this function will have zero angular density. If p = lim,_oo per) is an integer, then it follows from Theorem 18 of Chapter I that "
0,=
.
S,(r)
hm L-() =0. 1'-+00 r
Hence, by Theorems I and 2 of Chapter II, th~ functionf(z) has completely regular growth and its indicator is equal to zero. Thus by Theorem 5 we obtain the following result. COROLLARY l. within an angle (° 1 ,
If F(z) ( 2 ),
is a holomorphic function of proximate order per) while 'F(z) is an entire function such that .
M,(r)
hm --(-=0,
1'-+co r P '"
then the indicator of the product of thest' functions within the angle (° 1, 02) is equal to the indicator of the function F(z).
From Theorem 4 it is easy to obtain the following corollary. 2. If the functions fez) and 'F(z) are holomorphic within an angle if 'F(z) is of completely regular growth for some p(r), and if
COROLLARY
(° 1,
( 2 ),
lJ!'(z)
=
fez) ,(z)
is a holomorphic function within the same angle, then
To derive this equation one considers the product IP p.
The set of zeros of the entire function 00
cI>(z) =
II(I- ~z:) k=1
has an angular density. Furthermore, /S~ = 0. Thus (f)(z) is of completely regular growth. Let us mark the points r k lying in the intervals (22m - I , 22m) (m = 1,2, ... ). Next we renumber them in the order of increasing moduli, and denote them by {ap}. Analogously, we denote by {h p} the set of points rk lying in the int~rvals (22m, 22m +l), and we set
II(I - a,z:r ,=1 II(I + b,Z;:)3. 00
W(z)
=
00
,=1
The set of zeros of the function 'Y(z) does not have an angular density in any angle containing a ray of the set arg z = br/n (k = 0, I, ... , 2n - I), and therefore 'Y{z) is not a function of completely regular growth in anyone of these angles. However, on each ray of the type arg z = (2k + 1)'IT/2n(k = 0, I, ...• n - I) we have II¥(z) I = I cI> (z) I. and therefore the function 'Y(z) is of completely regular growth on each one of these rays. The function 'Y(z) thus serves as the required example. The situation is quite different if one requires additionally that the indicator of the growth of the function j(z) be sinusoidal within the angle [ex, P], i.e., if
"r(lj) =
AcoslJ +8 sin a
In this case, for the investigation of the behavior of the function within the angle [ex, Pl, we transform formula (3.04) by' integrating both sides of the equation
162
FUNCTIONS OF COMPLETELY REGULAR GROWTH
[CHAP. III
with respect toO from ffl to fP2' and with respect to 0 from 'f2 to '1:\ (IX q2 < q 3 < (j). Thus we obtain the equation
f
If.
'h
2-:rS "
NF(r, 1\, O)dfldlJ
~I
I, l'
=
< q1
+JF(!Pt)(!P3-~2)) t
o If,
to,
+ Sf '
