Research Not s in Mathematics
H JW Ziegler
Vector valued Nevanlinna Theory
Pitman Advanced Publishing Program BOSTON·...
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Research Not s in Mathematics
H JW Ziegler
Vector valued Nevanlinna Theory
Pitman Advanced Publishing Program BOSTON· LONDON · MELBOURNE
73
H JW Ziegler University of Siegen
Tvector valued Nevanlinna Theory
Pitman Advanced Publishing Program BOSTON· LONDON· MELBOURNE
PIlMAN BOOKS LIMITED 128 Long Acre, London WCZE 9AN PIlMAN PUBLISHING INC 1020 Plain Street, Marshfield, Massachusetts 02050
Associated Companies Pitman Publishing Pty Ltd, Melbourne Pitman Publishing New Zealand Ltd, Wellington Copp Clark Pitman, Toronto First published 1982
© H J W Ziegler 1982 AMS Subject Classifications: (main) 30, 3OC, 30D (subsidiary) 31,32,53 Library of Congress Cataloging in Publication Data Ziegler, H. J. W. (Hans J. W.) Vector valued Nevanlinna theory. (Research notes in mathematics; 73) Bibliography: p. Includes index. 1. Functions, Meromorphic. 2. Value distribution theory. 3. Nevanlinna theory. I. Title. II. Series. QA331.Z53 1982 515.9'82 82-13202 ISBN 0-273-68530-1 British Library Cataloguing in Publication Data Ziegler, H. J. W. Vector valued Nevanlinna theory.-(Research notes in mathematics; v. 73) 1. Functional analysis 2. Vector-valued measures I. Title II. Series 515.7 QA320 ISBN 0-273-08530-1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording and/or otherwise without the prior written permission of the publishers. The paperback edition of this book may not be lent, resold, hired out or otherwise disposed of by way of trade in any form of binding or cover other than that in which it is published, witbout the prior consent of tbe publishers. Reproduced and printed by photolithography in Great Britain by Biddies Ltd, Guildford
To RENATE
Contents
Preface
1 Extension of the First Main Theorem of Nevanlinna Theory and Interpretation by Hermitian Geometry
§ 1
Generalization of the Formula of Poisson-Jensen-Nevanlinna
1
§ 2
Interpretation by Hermitian Geometry
9
§ 3
The Generalized First Main Theorem
16
§ 4
The Example of Rational Vector Functions
27
2 Some Quantities arising in the Vector Valued Theory and their Relation
34
to N evanlinna Theory § 5
Properties of
V(r,a)
§ 6
Properties of
T(r ,f)
§ 7
The Connection of
§ 8
T(r,f.) ,m(r,f.) and J J The Order of Growth
34 39
T(r,f) ,m(r,f)
and
N(r,f)
with
N(r,f.) J
45
51
3 Generalization of the Ahlfors-Shimizu Characteristic and its Connection
58
with Hermitian Geometry
o
T(r,f)
58
§ 9
The Generalized Ahlfors-Shimizu Characteristic
§ 10
The Generalized Riemann Sphere
§ll
The Spherical Normal Form of the Generalized First Main Theorem 73
§12
The Mean Value Representation of
68
o
T(r,f)
75
82
4 Additional Results of the Elementary Theory
§13
The Genus of a Meromorphic Vector Function
§14
Some Relations between
M,m; N,n ; V,v
and
82 A
89 vii
5 Extension of the Second Main Theorem of Nevanlinna Theory
110
§15
The Generalized Second Main Theorem
110
§16
The Generalized Deficiency Relation
129
§17
Further Results about Deficiencies
141
Appendix: Rudiments of Complex Manifolds and Hermitian Geometry
168
Bibliography
189
Table of Symbols
197
Index
199
viii
Preface
= f (z) be a meromorphic function in the Gaussian complex plane C. 1 Nevanlinna theory or the theory of value distribution gives answers to the
Let
w
1
question of how densely the solutions of the equation (z E C ,01 1 E CU{oo})
are distributed over
C; it also studies the mean approximation of the func-
tion
f 1 (z)
to the value
a1
along large concentric circles around the ori-
gin
z =0
a problem which turns out to be equivalent to the former.
Nevanlinna theory originates from a general formula of F. and R. Ncvanlinna [45], by which they were developing a general method for the investigation of meromorphic functions. This formula includes both the Poisson formula and the Jensen formula as special cases, and in its most important form it expresses the logarithm of the modulus of an arbitrary meromorphic function by the boundary values of the function along a concentric circle around the origin and the zeros and poles of the function inside this circle. Nevanlinna theory was created at the moment when Rolf Nevanlinna gave the formula an ingenious interpretation, This happened about 1924. The most general result of Nevanlinna theory can be summarized by saying that the distribution of the solutions to the equation
f 1 (z)
= a1
is extremely uniform for almost all values of
a1 ;
there can only exist a small minority of values which the function takes relatively rarely. The investigation of these exceptional values constitutes the main task of value distribution theory in the sense of Nevanlinna. The earlier value distribution theory before Nevanlinna can be traced back to the year 1876, when K.Weierstrass [57] showed that in the vicinity of an isolated essential singularity a meromorphic function given value
a1
f 1 (z)
approaches every
arbitrarily closely. In 1879 E.Picard [50] even proved the
surprising fact that a meromorphic function takes in the vicinity of an isolated essential singularity every finite or infinite value
a1
with 2 exceptions at
the most. Points which are not taken are now called Picard exceptional values of the function. The results which were found after by the mathematicians E. Laguerre, H.Poincare, J.Hadamard, E.Borel and others revealed that in spite ix
of the possible existence of Picard exceptional values the distribution of zeros or, more generally, the distribution of a -points of an entire function is controlled, at least in some sen se, by the growth behaviour of the maximum modulus function max !f 1 (z) I Izl= r which has the function of a transcendental analogue of the degree of a polynomial. This approach of early value distrib ution theory breaks down, however, if
f 1 (z)
is meromorphic, since then
has a pole on the circle
M(r,t 1 )
becomes infinite if
f 1 (z)
Izl = r . An attempt by E.Borel [3J himself of in-
cluding meromorphic functions in this framework was not very successful. In Nevanlinna theory the role of valued function
logM(r,f 1 )
is taken by an, increasing real
T(r, f 1) , the "Nevanlinna characteristic function" which is
associated to the given meromorphic function
f 1 (z) . A great deal of work
had been done in establishing the relationship between distribution of values and growth when Rolf Nevanlinna created his epoque making theory. This theory, which applies to entire functions, as well as to meromorphic functions, even improved tremendously the earlier value distribution theory of entire functions. There have been many attempts to extend the Nevanlinna theory in several directions. Besides older investigations of E.Borel, A.Bloch, R.Nevanlinna and H.Cartan the most important of these, known as the theory of holomorphic or meromorphic curves, was initiated by H. and J.Weyl [43] in 1938; the most difficult problem of this extension, the proof of the defect relation for holomorphic curves, was solved by L.Ahlfors [2] ; recently a very modern treatment of this theory was given by H. Wu [46 J • In its most simple form this theory investigates the distribution of the zeros of linear combinations A f (z) + •.• + A f (z) 00 nn
of finitely many integral functions stan t multipliers
A = (A
0'
... ,A) n
w. = f.(z) for different systems of conJ ] or, in other words, this theory analy zes
the position of a non-degenerate meromorphic curve
C -
pn
relative to
the hyperplanes
A w + ... + A w = 0 in complex projective space pn. o 0 n n The theory of holomorphic curves by Weyl-Ahlfors was further extended in a very general way to a higher dimensional theory first by W. Stoll [56] and then in a different direction, stressing Hermitian differential geometric aspects, by
x
H.I.Levine [43], S.S.Chern nO], R.Bott and S.S.Chern [7] and other authors. In 1972, introducing once more fascinating new ideas, the Ahlfors-Weyl theory was extended in a different direction, more regarding to algebraic ge ometry, by J . Carlson and P . Griffiths [IS] to equidimensional holomorphic mappings Cm_ V ,where V is a projective algebraic variety and where
m
m
you are interested in how the image meets the divisors on
V
m
. This theory
was further generalized in the same direction by P.Griffiths and J.King [30] to the study of holomorphic mappings A-V
f
where
A
is an algebraic,
braic subvariety
Z
C
V
V
a projective algebraic variety. Given an alge-
the 2 basic questions which are treated in this
setting are in analogy to Nevanlinna theory: (A) can you find an upper bound on the size of
CI(Z)
in terms of
Z
and the "growth" of the mapping f ;
(B) can you find a lower bound on the size of
f-I(Z)
,again in terms of
Z
and the growth of the mapping. The most important special case of this problem is when A::: C m and V::: pn ,the complex projective space. Then f may be given by n
meromorphic functions Z ::: (z
The subvarieties Pc/wI"" ,w n ) the equations
Z
l'
.. '
z
'm
) E Cm
will be the zero sets of collections of polynomials
and so the questions amount to globally studying solutions to
Concerning the extensions of Nevanlinna's theory for holomorphic mappings between Riemann surfaces, we refer to L.Sario and K . Noshiro [30] and to the more Hermitian differential geometrical versions of S.S.Chern [11] and H.Wu [62] . We disregard here the several extensions of Nevanlinna theory to certain classes of non-holomorphic functions, note however the extension of E.F .Beckenbach and G.A . Hutchison [4] to triples of conjugate real harmonic functions, as it rests on a similar underlying idea as our developments. In this Research Note I am presenting an extensi9n of the formalism of NeyanIinna theory to systems of
... ,f (z) n
n
~
1
meromorphic functions
in a way, which is fundamentally different from the theory of
holomorphic or meromorphic curves of Weyl-Ahlfors and its higher dimensional
xi
generalizations. As in Nevanlinna theory again the starting point is a formula, this time a generalization of the formula of Poisson-Jensen-Nevanlinna, which I discovered in 1964, when I was trying to extend the Nevanlinna formalism to the simultaneous solutions of systems of n
equations
=
f (z) n
=
a
n
z E C
a
n
E
C
w. = f.(z) ,j=l,···,n are n:c I meromorphic functions. J J We note that already G.P61ya [52] and R.Nevanlinna [47] have studied func-
where
tions with respect to values they assume at the same points; contrary to the present study, however, these values were taken to be one and the same complex number
a 1 • and they investigated the condition under which necessarily
fl(z) == f 2 (z) I succeeded in extending formally both the main theorems of Nevanlinna theory. together with the Nevanlinna deficiency relation. Although the above system of equations has only solutions for points set which is rather thin for
a = (a l ,··· ,an) E f(C) a n > I " these results seemed to be quite inter-
esting. However, one difficult main problem was stilI to solve; the problem of finding the true geometric meaning of the extended quantities. a problem which was proposed to me by Helmut Grunsky and by Rolf Nevanlinna in 1967/68. It took me several years to find its solution. and was finally achieved partly in my doctoral thesis and partly in my Habilitationsschrift, on which the present Research Note is based. The main difficulty was the appearence of a new term to the generalized Nevanlinna value distribution quantities and
T(r,£)
V(r,a)
in addition
m(r,a), N(r,a)
in both the extended First and Second Main Theorems. I then
tried to compare my result with the totally different theory of Weyl-Ahlfors and its extensions and with the more recent developments of complex manifolds and Hermitian geometry. So I gradually found out that the notions of the m dimensional complex projective space
pm
,its Fubini-Study Kahler metric and the
complex differential geometry of Hermitian line bundles, which are now a central fact in the theory of holomorphic or meromorphic curves and in generalized multidimensional Nevanlinna-Weyl-Ahlfors theories, are also the key for the
xii
proper understanding of the geometric meaning of the main new term and even for the interpretation of my tic function"
o
II
V(r,a),
generalized Ahlfors-Shimizu characteris-
T(r, f) . A fundamental role is played by the curvature form,
whose cohomology class represents the characteristic Chern class of the hyperplane section bundle over complex projective space. While the theory of holomorphic curves by Weyl-Ahlfors and its generaliza-
tion give results for the problem of how often the image of the mapping meets a set of hyperplanes or subvarieties in the image space, the theory of the present Research Note does not render results in this direction when
n
~
2 . In
contrast to this we study the growth of the projection of the curve f(z) - a
C -
Cn
into
pn- 1
and the connection of this growth with the distribution in
C
of the solutions
to the system of equations f( z)
whilst the point counterpart for
a
a
varies over
Cn
,a problem which has no effective
n = 1 . The theory reduces to Nevanlinna theory if
and stays in close contact with the original Nevanlinna formalism if
n = 1 n
~
1 .
The readers of this book must have hardly any prerequisites from Nevanlinna theory, but only a good lesson in function theory. An advantage. however - even a necessity in some places - would be a little familiarity with the sources R.Nevanlinna [46], [27] or [28]. In Chapter 1, §2 , Chapter 3, §9 and in Chapter 5, §15 some background knowledge of fundamental ideas from Hermitian geometry and algebraic geometry might aid the deeper understanding of certain formulas. The interested reader will find some hints on these matters in the Appendix and further details in the literature of the Bibliography. With pleasure I thank W.Helmrath and Professor R.Schark for correcting part of the manuscript, but above all I want to thank my wife for helping me with the literature and having all the patience during the time of writing. Last but not least, I would like to thank Pitman Publishing for their excellent cooperation.
Siegen, West Germany
Hans J. W. Ziegler
April 1982 xiii
1 Extension of the first main theorem of N evanlinna Theory and interpretation by Hermitian geometry §1
Generalization of the formula of Poisson-Jensen-Nevanlinna
We denote by
en
the coordinates
the usual w
n
dimensional complex Euclidean space with
=
the Hermitian scalar product
+v w
n
n
and the distance
1
Ilv - wll
=
+
2
Let
= be
n
~
•••
J
W
=
n
f (z) n
1
complex valued functions of the complex variable meromorphic and not all constant in the Gaussian plane e 1
z ==
,which are e, or in a
finite disc
{ Iz I
=
0
1
~
w)
v
is
dT
f* ( 1
2'IT f*
1\
a2 log 1[£(011 d s a~
'IT
-1fi
where
da
f
'IT
.l'IT f* W
=
. By Wirtinger's theorem,
I u m e e l e men t of the curve
f,
pulled back to
CR'
v(r,O)
j
1
=
-
1T
C v(r,D)
then
2i1
C
v
IIf(UI!
do /\ dT (2.4)
I ume
0
(the volume is 1Tv(r,O»
of the holomorphic curve
r
is due to the fact that if tive subspace of pn-l then
The factor
/:, log
cr
r
is the normali zed
of the restriction to
j
1
f * u)
is anyone-dimensional projec-
1[
1T
In the N evanlinna case is a point, and
v(r, 0)
n = 1
f
reduces to a constant map since
pO
vanishes identically in this case.
In concluding this section we note for later reference the following explicit formula, which is obtained by the computation (2.2) or by the direct computation given below:
1 21T !1 log
IlfUJl1
1 f* 1T =
=
i 2'/T
i 2IT
do /\ dT
W
dE.; /\ d'"[
2
1
IifIi4
L
If/k - fkfjl2
d~
(2.5)
/\ d'"[
j < k
outside the exceptional points, where
f(
~)
vanishes or has a pole. Here the
last line comes from
2
L j < k
If/k - fkfjl2
=
L j,k
U/k - fkfj)(I{k -
rk~)
=
=
fl [Crk'
f.fl'rkc J ~ J
~
+
JJ
- - ]
It is easy to see that (2. 5) can be con tin uously extended in to the exceptional
points. From (2.5) we see in particular that the integrand in (2.4) is
~O
The direct computation of (2.5) runs as follows. Using the relations
= for holomorphic d dO log
=
A, we obtain
1
I[ f( t;) [[
2 =
1
d
ao
log
+
Re
2
-i(- ] 2
log
[[f(OI[
= 1m
log [I f( 0
2 + 1m
II
2 [1m
p
choose a number
].I
p.
J such that
p
>].1
> max Pj
. From the right
inequality (8.4) and Proposition 8.3 we conclude +00
n
jL
k=l
T(t, f k ) t
].I
+1
dt
=
+00
to
> 0
,
to which is impossible. since by the same Proposition each integral
55
+00
I
k=l, ... ,n
is convergent. Thus we must have
max . J
this proves (8.2).
, and observing (8.4)
p.
J
(8.3) follows immediately from the left hand inequality (8.4).
Let
f
o
-+oo
T(r,f.) J rP
+00
p
-+oo
~
T(r,f k )
L
lim r->-+oo k=l·
~
rP
rP
(j = 1, ... ,n) . Here in the extreme left and right hand sides all terms vanish which belong to component functions of order are the component functions of order
P
. Thus if
f. , . . . ,f. J1 Jm , then we have the following ine< P
qualities between types m ~
T.
J.1
i=l
In particular, if
f. , . ) 1 If
f
L
~
T
f
is of order
(8.6)
T.
Ji
o
0
to
t =r
(r O < r < R)
we get
r
J 2~: J Alogv'" 1'0
112 dx
11£(,)
, dy
Ct
r
21f
J n(t,f)~n(O,f)
dt
r + n(O,f)log-
+
rO
1 2n
JlogY/I
+ II[(re I"e )11 2 de
o
rO
(9.3)
At
Z :::
0
we have the vectorial Laurent-development + z q +l c
f(z)
q+l
q ::: n(O,O,f) - n(O,f)
where
ros and poles of
f
at
is the difference between the number of ze0
Z :::
+ .• "
,and
IIc q II
'"
0
Concerning the development of
G) at
Z
:::
0
(column vector)
we distinguish three cases:
i)
q>
ii)
q:::O
0
iii) q< 0:
C) C) C)
:::
:::
:::
(~)
+
Go)
+
,q
zq(:q) +
zCJ +
CJ·
zq+l
z2
zq+l
GJ
(~
) q+l
+ .•.
+...
C
(0c q +l ) +... + Co )+
z
e)+.". c1
In these three cases respectively the expression
59
= behaves as
r0
-+
0
like
or
In the cases i) and ii) the point iii) the point " . ,0)
z:::
0
z::: 0
is not a pole of
f
In case i) we have
is a pole of
=
; in case ii) we have
1If(0)1!
.
f; in the case f(O) = 0 ::: (0,"
i) and
ii) can be
considered together so that we have logJ1 + IIf(0)1[2 finite at :::
)
Thus, if
z::: 0
l'~;
I
rO
Ct
r
=0
is not a pole of
'10g"\ +
lI£(z)II'dx
is
(cases i) and
ii) )
A
dy
f
is a pole of
lie q II f
+ o(r O)
if
(case iii»
we obtain from (9.3)
"
I 2')]
r
r
f
-n ( 0, f) log r 0 + log
z
if
+ o(r 0)
z ::: 0
n ( t , f)
~n ( 0 ,f) d
J
+
1 271
"e 2 de - logYI 1 + 11£(0)11 2 logYI 1 + IIf(re1)1[
o
rO so that, letting
t
r0
tend to zero,
-o(r o)
,
r
J 2~'tJ .'og/,+ 11£(')1I o
2 dxA dy
"
Ct
211 N (r, f)
+
J logYI 1
1 21T
II.
+ lIf(re l"e ) 2de
(9.4)
o If on the other hand
z::: 0
is a pole of
f
,then we get from (9. 3)
r
f 2~~ J .lDg~, rO
+
11£(,)11 2 dx
A
dy
Ct
211
r
J n(,.Ot(o.n d'
+ n(O,f)logr
1 + 21T
and letting
J
J logVl/
+ lIf(re l"6 )11 2 de
o
rO
r
"
r 0 -+ 0
2~~ J ",ogJ,
o
+
lIf{z)" 2 dx
A
=
dy
Ct 21T
N(r,f)
1 1T
r logYI 1 +
+ -2
J
"e 2 de IIf(rel)1I
-
log
IIcq II .
(9.5)
o Now the function
61
which appears in (9.4) and (9.5). behaves asymptotically very similar to mer,£)
; it can serve equally well as
proximation of
f
mer,£)
to infinity on circles
dC
as a measure for the mean ap. We can therefore introduce
r
the following modified proximity function
o
o
m(r,oo,f)
m(r,f)
with respect to infinity:
Definition 9. 1 :
21T
o
mer,£)
::
1 21T
f.j log
1 + \\f(re 1"e )11 2dE)
(9.6)
o if
Z ::
0
is not a pole of
f, and
21T
o
m(r,f)
::
,'. f 10gJ,
lIf(re")1I 2de
+
-
log
Ilc q II
(9.7)
o if
z:: 0
f
is a pole of
. Here
c
q cient vector in the Laurent development of
is the first non-vanishing coeffif( z)
at
Z ::
0
.
With this definition (9.4) and (9.5) can be written in this unified form: r
~(r,£)
+
N(r,f)
0
f 2~~ f o
AlogJI +
11f(,)1I 2dx
A
Ct
Now we already observed on p. 26 that in the scalar case
62
dy
.
(9.8)
and that
i JAlOg.,!'
+
1',(,)1' dx
dy
A
Ct is the spherical area of the Riemannian image of
Ct
under the mapping
f 1;
and in this case the sum in the left hand side of (9.8) is called the characteris tic function of AhHors and Shimizu or the spherical characteristic of
t
fl.
We now try to find in the present vector valued case the correct geometric meaning of the expression
i J . log.,!,
+
II f(,) II' dx
A
dy
Ct occurring in equation (9.8). We compute
i
tdog V\
i·4.
+ IIf( z) 112 dx
A
32 --10gJ1 + /If(z)11 2 dZd
z
=
i
( 1 + LkfkTk )( Lkfilk) -
"2
(1
=
dy
~ dz ~
A
dz
aalog(l
=
+
r.f{;
(Lkfkfk )( Lkfkfk )
dz
=
A
dz
(9.9)
+ Lkfkfk)2
tAhlfors [1]. Shimizu [54]
63
k = 1
Here each sum is extended from en
Now en £; pn
is sitting in
pn
to
as an open set, and we have the inclusion map
Pulling back by the inclusion we obtain the
II
k=n
the Fubini-Study metric on
F ubi n i - Stu d y
met ric
0
n
en
1\
-r.
pn
,
It is given
by ds 2 =
(1 + LkWkw k ) (Lkdw k 0 dwk"j- (LkWkdw k )0 (LkWkdw k ) ( 1 + Lk w~wk )2 (9.10)
Its Kahler form is
(1
+ LkWkw k )( Lkdw k
1\
dW k )-( LkWkdw k )
(1 + LkWkwk
= If
ds 2
"41 dd C log ( n ::; 1
-
1 + Lk ~ ~
1\
(
LkWkdw k )
l
)
(9.11)
for example, then
pI
is the usual spherical metric on
is the Riemannian 2-sphere
S2, and
e
dW10 dW1 - 2 (1 + wl~) which is a conformal (Hermitian) metric of constant Gaussian curvature
4
and the associated Kahler form dU l
=
1\
dV 1
::;
- 2 (1 + w 1w1 )
is the spherical volume form. We now return to the general case the pull-back of
64
to
is
n
~l
. (9.11) and (9.9) show that
*
( 1 + rkfkfk )( rkdfk " df k )-( rkfkdf k ) " (rkfkdfk ) i = u 2"
f w...
( 1 + rkfkfk
l
=
(9.12)
This shows that in the general vector valued case
i J AlogV'
+
11£(,)11' dx
~
1
the integral
=
dy
A
n
(9.13)
Ct is nothing but the vol u m e of the image of
(If (9.13) is divided by
Tr
in
Ct
pn
under the map
we obtain the nor mal i zed
vol u me).
We now return to equation (9.8). At this point, in view of equation (9.8) and the given geometric interpretation (9.13) of its right hand side, it is natural to introduce a new modified characteristic function
o
Definition 9.2
T(r,f)
=
o
m(r,f)
o
T (r, f)
by
+ N(r,f)
(9.14)
o where
mer,£)
We will call z u
is given in Definition 9.1 .
o
T (r ,f)
the" g e n era 1 i zed
c h a r act e r i s tic
II
or the
II
A h 1 for s
g e n era 1 i zed
- S him i -
s p her i cal
c h a r act e r i s tic ", since it agrees with the characteristic of AhHors Shimizu in the scalar case. Summarizing, we can formulate this result: Theorem 9. 3:
Let
Then denoting by en
f
uu
be a vector valued meromorphic function on C R the Kahler form (9.11) of the Fubini-Study metric on
,and defining the generalized spherical characteristic
o
.
T(r,f)
by Defi-
nition 9.2 , we have the formula:
65
r
o
T (r, £)
::
2~ j ~t
j
o
Ct
'10g/1 ,
,
o
This geometric interpretation of
T (r, f)
terpretation of the generalized characteristic
Ilf(,)11 2 dx
, dy
(f '" con st.)
(9.15)
.
gives also a quasi-geometric inof N evanIinna because
T (r , f)
of the following Proposition 9.4
T(r,f)
o
T(r,f) as
r
-+
R
T (r, £)
::
o
differs from
T(r,£)
only by a bounded term:
(9.16)
O( 1)
+
.
Thus, in many investigations
o
T
ean be used instead of
T
and vice
versa, without any changes of formulas. The estimate (9.16) can be seen as follows. We have
211 0
m(r, f)
::
21,
j
10g/1 , Ilf(re ie ) 112 de
d
0 where d if
::
z :: 0
pole for
logJ1 + Ilf(0)11 2 is not a pole for
f
16g Ilfll
, and
f
d
::
z :: 0
if
log lie II q
Since
1
therefore, it
vanishes up to a set of 2n - dimensional measure zero on the sphere
S2n
.
Further,
i
i~t i r
V(r ,a) dS'"
o
S2n
v(t,a) dS 2n
S2n
and we know that this vanishes identically for (12.2) can be written
+
Is nl
2n V(r,a) dS
. Using these facts,
n > 1 ,
if
S2n
0
T(r,f)
i
n = 1
=
l11
i
(12.3) N(r,a) dS 2
n = 1
if
s2 If we introduce the abbreviation
A(t,f)
1 -2 Is nl
i
by putting
v(t,a) dS 2n
if
n > 1
S2n A(t,f) if
n
=:
1
S2
77
then (12. 3) can be written
j r
o
T(r,£)::
A(t,O
dt
t
o in the case
n:: 1
this is a well-known formula of Nevanlinna theory. We
summarize this result in Theorem 12.1:
o
The generalized spherical characteristic
constant vector valued meromorphic function in
CR
T(r,£)
of a non-
can be represented by
the integral mean
if
o
T(r,£)
n > 1
,
(12.4)
::
~f
N(r,a) dS 2
n ::: 1
if
S2 Equivalent to this is the formula r
o
T (r, £)
f
:::
A(t,£) ---dt
(12.5)
t
o where
A(t,£)
denotes the spherical mean given by
1 f Is 2n l
v(t,a) dS 2n if n > 1
S2n A (t, f)
1 f 0
the integral
+00
j
T(r,f) r
fl+1
dr
rO > 0
rO is convergent, then the same holds for the integrals
j
V(r,a) - - ' 1 dr r fl +
j
+
logM(r,a) - - - - ; - - dr r
(14.15)
fl+1
and the series
1
Iz. (a) Ifl Iz.(a)l> 0
J
J
Proof. The convergence of the series follows from Proposition 14.4, observing that by the first main theorem the integral
j
N(r,a) r
fl+1
dr
is convergent. The convergence of the left integral (14.15) follows from the first main theorem. The statement that the right integral (14.15) is convergent follows from Lemma 14.9 and Theorem 14.3, in the case
a E en
by combining
the inequalities (14.9), (7.19) and the footnote on p. 97. This proves Proposition 14.11. Proposition 14.12:
Let
morphic vector function. If for given
102
be a non-constant mero].I
>
0
the integrals
+co
+co
+
log M(r,a) -----;-- dr
f
r
V(r,a)
)1+1
f
r
1l+1
(14.16)
dr
and the series (14.17)
are convergent for some value a E C n U {co} ,then they stay convergent for every a E C n U {oo} ,as does the integral +co
T{r,f)
f Proof.
r
)1+1
Since
( 14.18)
dr
m(r,a)
is majorized by
+
logM(r,a)
,the integral
+00
m(r,a)
f
r
1l+1
dr
rO is convergent for the given value of
a
. Since the series (14.17) is conver-
gent, also the integral +00
N(r,a) ---=--1 dr
f
rll +
is convergent by Proposition 14.4. Observing that the right integral (14.16) is convergent, we obtain from the first fundamental theorem that the integral (14.18) is convergent. The convergence of (14.16) - (14.17) for every
a
follows from Proposition 14.1l. Combining Lemma 14.9, Proposition 14.11 and Proposition 14.12 we have
103
Theorem 14.13:
Let
fez)
phic vector function. Let
= (f 1 (z), ... ,fn (z»
be a non-constant meromor-
denote the order of the mean value
01 (a)
r
~f
'6gM(t,a) dt
o denote the limit inferior of the exponents
Let
°>
,for which
0
the series 1
Ida)!> J
a
converges. Let
v(r,a)
denote the order of
(or V(r,a»
. Then
o(a) rests in variable for all
a E en
u
and is equal to the order
{oo}
of f( z).
p
From the generalized first main theorem + N(r,a)
m(r,a)
=
+ V(r,a)
T(r,f)
+ 00)
we obtain (
m(r,a)
lim r++ oo
N(r,a) +
+
T(r,f)
V(r,a) )
=
(14.19)
1
T(r,f)
T(r,f)
which shows that the limit inferior and the limit superior of each of the quo tients N(r,a)
m(r,a)
VCr ,a) and
T (r, f)
lies in the closed interval
n
m(r,a)
since
We also remark the following:
and
N(r,a)
min {p l' ... , p n }
are of order
is of order
p , or
,then for
and
, then by the first main theorem T(r,f)
f = (£ l' ... , fn)
is the order of
p
If
104
[0,1]
= max {p l' ... , p n } non-constant) and if p* = If
f
T(r,f)
T(r ,f)
a=oo
V(r,a)
V(r,a)
(£ l' ... ,
a E en ~
p.
must be of order
p,
is of order
Lemma 14.14: Let the functions
o
B
0
f( w)
f( 0)
"
on
for
Jwl
Jw! = 1'0
04.24) then
1'0
it follows that
[£(w) ,£(0)] -
[£(0) ,a)
so that by definition of
B -
":
!
B
=
!
B
o
m (Definition 11.1),
Hayman [l6],p.15
107
I 2'Tf
o
~ 2~
0
- m(r,a) + m(rO,a)
log
2
log 6
lie de [f(roe ) ,a)
o Thus in this case we have using (14.24)
I r
o
v(L,a) +t n(t,a) dt
= V(r,a)+N(r,a)-V(ro,a)-N(ro,a) < T(r,f) +log
2
i3'
(14.25)
[f(O),a]
On the other hand, if
1 log---[f(O),a)
then
2 S
log -
o
m,
and, using (14.23) and again the definition of
I r
v(t,a) +t n(t,a) dt
Thus 04.25) holds for
and all
a
o 2 T(r,f) + logS
~
V(r,a) + N(r,a)
log r 0
and the second function is strictly increasing. Thus by Lemma 14.1S we can find a sequence that for
r . .". +00
depending on
]
r = r. ]
and all
o
T(r,f)
a
d r dr [V(r,a)+N(r,a)]
0
will be called the ffd e f i c i e n -
131
~If
of the quotient V(r,a)
N(r,a)
+
T(r ,f) or simply of the point
0 (a)
with
0
>
will be called the
a
,quite analogous to Nevanlinna theory. Points
will be called
"d e f i c i e n t
of m u I tip I i c i t Y
"i n d e x
is positive only if there are relatively many multiple by a multiple tion s
f ( z)
The quantity
If.
of
TI
a
e(a)
since
a - points of
a
e(a)
f
; here
a - point we understand a point such that the system of equa-
=a
f' ( z)
has multiple roots; these roots are zeros or poles of
and are thus countable in number.
e(a)
1
will attain its maximum
if the
relative density of multiple roots is large, and if their orders of multiplicity are z =
unbounded in the vicinity of 0 (a)
deficiency
o (a)
+
00
•
e(a) = I
For such a point with
the
must vanish since
e( a)
:;;
1
Remark. If we define as in Nevanlinna theory the number
nI(t,a)
by
setting n(t,a) nI(t,a) f( z)
-
n(t,a)
is the number of multiple solutions in
= a
,where a solution of multiplicity
We can then introduce the
TI
\)
Izl:;; t
of the equation
is counted only
(\) - 1) times.
counting function of multiple a -points
by putting
I
If
N I (r, a)
r
nI(t,a) - nl(O,a) --------dt
N(r,a) - N(r,a)
t
o for fixed
r
there are only finitely many
we have
=
L
NI(r,a)
a E Cnu {co} so that
132
a
for which
N I (r, a) '1= 0
,and
N 1 (r) lim - - r-+R T(r,f)
N 1 (r,a) lim - - - - r-+R T(r,f)
L
a E en u {oo}
Now, as an essentially new ingredient,as compared to Nevanlinna theory,we introduce the quantity
the resulting sequence and putting
q
:3 '
{a
etc .. If
a (0)
=
00
,
(fI.)
}
_ (fI. - 1, 2, ... )
is
we deduce that
2
+
for any finite
1
,and hence if the sequence
{a (fl.)}
is infinite we con -
elude that +00
2
+
This proves Theorem 16. 1. Using the definition of
e( (0)
the right hand inequality of the deficiency
relation (16.14) can be written
L
e(a)
+
1
N(r,I) lim - - r+R T(r,I)
+
2
0G
with
where now the sum is to be taken only over any set of >
Sea)
0 • This gives with (16.9)
L
[0 (a) + 6(a)]
L
N(r,£) flea)
:;;;
1 + lim
r+R T(r,I) ( 16.15)
We deduce in particular Corollary 16.2 : If under the conditions of Theorem 16.1 the meromorphic vec-
136
tor function
f
has no poles in
N(r, f) lim r-+R T(r,f)
or more generally, if only
o 2
then the number
(16.16)
on the right side of the deficiency relation (16.14) can
1
be replaced by
CR
provided that the sums are extended over finite
a
only,
i.e.
L
[0 (a) + e(a)]
L
+
9(a)
( 16.17)
+
a E Cn For example (16. 17) is always satisfied for an entire vector function. From (16.15) we see further If
Corollary 16.3: in
CR
0
,then the
:;;
f( z) is an admissible vector valued meromorph.ic function inde~
N(r,f) 1 + lim r-+R T(r,f)
:;;
°G
of the Gauss map satisfies the inequality
( 16.18)
In particular, 0 if
f( z)
0 if
:;;
:;;
°G
2
is meromorp hic ; and :;;
f( z)
:;; 1
°G
is holomorp hic .
In the sum m(r,a) the terms
+ N (r, a)
+ V(r,a)
m(r, a)
and
V(r,a) + N(r,a)
tically very differently as convergence of
f(z)
to
weak, so that the deficiency
r -+ R a
behave for fixed
. For the "normal" points
,which is measured by ° (a)
m(r,a)
a a
asymptothe mean
,is relatively
will vanish, i.e.
137
m(r,a) lim r-+R T(r ,f)
"-
0
whereas V(r,a) + N(r,a) lim r-+R If
I)
(a)
1 T(r,f) V(r,a) + N(r,a)
is positive, then the growth of
weak; in general the number V(r,a)
I)
(a)
is relatively
is a measure for the growth of the sum
+ N(r,a)
If in particular for a transcendent meromorphic vector function in
point
a
C
the
is a generalized Picard exceptional value, then the deficiency
attains its maximum value
1
=
logr
since
o(T(r,f)
0 (a)
. Therefore
Theorem 16.1 contains the generalized theorem of Picard (Corollary 15.5), ac-
2
cording to which there are at most
distinct such values
a
On the other hand, for a meromorphic vector function in
C
o (a) = 1
the equality
which is equivalent to V(r,a)
lim r-+ +00
+
N(r,a)
o
T(r,f)
does by no means imply that the point
a
is a generalized Picard exceptional
value. The notion of deficiency allows to distinguish between possibly countable infinitely many exceptional points as compared to only
2
such values in the
generalized Theorem of Picard-Borel. From Theorem 16.1 we also note that there are at most a
2
distinct points
for which
2 - 0 I)
(a)
>
G
3 and in particular at most
2
points with
Corollary 16.2 we see that there is at most
o(a)
o(a)
1
2
>
"3
analogously from
finite point
a
o (a)
>
such that
1 - 0G >
2 and in particular at most
1
One can pose the general
138
finite point such that
1
"2
Problem:
Given sequences
°G ;:
and a number
0
,such that
f(z) ::: (f1(z), .. ·.,fn (z»
is there a vector valued meromorphic function 51,
::: e51,
o(a )
°
and with
G
the Ricci-index of
,
f( z)
and
o(a) ::: 6(a) ::: 0
for
a
with
¢
{a 9-} ,
?
In scalar NevanIinna theory this deep problem, of which only partial solutions have been known earlier, has recently been given a positive answer t We now introduce an important new concept, which has no significant counterpart in scalar Nevanlinna theory, by setting
I
:::
Y(r ,a) lim r+R T(r ,f)
-
m(r,a) + N(r,a) ( 16.19)
lim
:::
T(1' ,f)
In view of the first main theorem we have always
and in particular
('\ Y(
where as above
o a
z ___ co
For an entire rational vector function n ( +00 , a) -
L a
E
this simplifies to
n(+00 , a) +
(17.10)
1
*( 00 )
en
n ( +00 , a) >
Example 1.
n(+00 , a) As an example we consider the entire rational vector function
3 , z 5)
f( z)
(z
Here every point
a E
e2
not of the form zo *- 0
not assumed. In every point
(z~,z~)-pointofmultiPlicity point of multiplicity
3
We have
5
where
f( z)
*(00)::
the function
l,andin
fez)
zO::O
2:0
has an
it has an
n(r,a):: n(r,a) n( +00 ,a)
1
:::
up to the point
2
1
+
f' (z)
(3z
N(r,O,f')
:::
4
, 5z )
2logr
3
= 1 ,
=
B( co
)
0
,
= 0
N (r, f)
=
,
0
B( co
= lim
)
r-+R T(r,f)
N (r, f) - N(r,f) ( iii)
B( 00
) = 1
lim r-rR
=>
1
e( 00)
N(r ,f)
N (r, f)
0(00)
= 0
, lim
= 0 , lim
r-rR T(r,£)
(iv)
(v)
146
0(00)
e( 00)
= 0
= 0
= 1
T(r,f)
=>
N(r,£) lim r-rR T(r ,f)
= 1
=>
N(r,f) lim r-rR T(r,f)
= 1 ,
0(00)
= 1.
r-rR T(r,f)
= 0 ,
B( co )
= O.
( vi)
e( 00 )
N(r,a) lim - - r+R T(r,f)
= 0
:;;
N(r,a) lim - - r+R T(r,f)
We also note Proposition 17.2
be a non-constant mero-
Let
morphic vector function in
CR
. Then the following inequalities are valid for
aECnU{oo} N(r,a) lim r+R T(r,f)
N(r,a) lim r+R T(r ,f)
(17.11)
N(r, a)
N(r,a) lim - - r+R T(r,f)
(17.12)
lim r+R T(r,f)
Proof of (17. ll) . m(r, a) lim - - r+R T(r,f)
m(r,a) + N(r,a) lim r+R
T(r, f)
- m(r,a)
m(r,a) + N(r,a)
=
+
lim T(r,f)
r+R
T(r,f)
The right side is :;;
N(r,a) lim r+R T(r,f)
~
and
N(r,a) lim r+R T(r,f)
This shows (17 .ll); (17.12) is shown analogously. From Proposition 17. 2, from the inequality
e(a) + 6 (a)
:;;
or from
Sea)
the definitions we deduce Proposition 17.3:
Let
morphic vector function in
fez) = (f 1 (z), ... ,fn (z»
CR
VCr ,a)
0)
6 (a)
1
be a non-constant mero-
. Then the following conclusions 'hold.
lim r+R T(r,f)
=
0 Sea)
N(r ,a) lim - - r+R T(r ,f)
=
1,,' e(a)
=
=
0
,
0
147
48
N(r,a) Oi)
o (a)
::::
0
N(r,a)
lim
'*
~
lim
:5
6 yea)
lim r .... R T(r,f)
r .... R T(r,f) N(r,a) ( vi)
lim
---
;:
0
oyea)
=>
(:I( a)
r .... R T(r,f) 6(a)
( vii)
6(a)
1
=>
B(a)
;:
;:
N(r,a) lim r+R T(r,f) VCr ,a) lim r-}-R T(r,f)
1
N(r,a) lim - - r .... R T(r ,f)
;:
V(r,a)
lim
( viii)
0
0
;:
6 yea)
;:
O(a)
;:
1
o
r .... R T(r, f)
o (a)
N(r,a) ;: I-lim - - - , r .... RT(r,f)
N(r ,a) 0)(a) ;: I-lim - - -
r-.. RT(r,f)
0
(ix)
1
If
(17.13)
,then
have
2n - dimensional Lebesgue measure zero since these sets are subsets of f(" C). If
relation
n:O; 1
is admissible and if
f( z)
E(,),
and
E( 0)
,then by the generalized deficiency
can be at most countable.
E(e)
From (17.13) and the deficiency relation we deduce If
Proposition 17.7:
is admissible and if
f( z)
is countable. On the other hand, if
E(6 V )
is countable, then
E(
I)
N)
f(z)
E( 1I N)
is countable, then
is admissible and if E(6 V )
is countable.
If condition (ii) of Lemma 17.5 is satisfied, then condition
0) is also satis-
fied and we deduce from Lemma 17.5 and the deficiency relation Proposition 17.8:
Let
Assume that for all
f( z)
a E en
be a meromorphic vector function in the plane. one of the following 2 conditions is satisfied:
N(r,a)
o
lim r->-+oo T(r,£)
(ii) Then the set
f( z)
6 V (a)
6(a)
E(
is transcendent and
I)
V)
::
E(E-)
::
EJ(a)
:: E(6)
e(a)
n ( +00 , a) ->- +00 ::
0
holds for all
is countable, and the deficiency rela-
151
tion can be written in any of the following 3 identical formulations:
I:
(i)
°G
+
°V(a)
~
1
+
N(r,f) lim r++ooT(r,f)
~
1
+
N(r,f) lim r->-+ooT(r,f)
1
+
N(r,£) lim r-+ +00 T (r , f)
a E en
I: °(a)
(ii)
+
°G
+
°G
a E en
I:
( iii)
Sea)
a Ee n
~
Assumption 0) of this proposition means that there are not too
Remark.
many a - points for all finite
a
; this will frequently be the case. One can
ask on the other hand what happens if there are many a - points for some particular
a
1 -
. From the first main theorem we obtain the inequality V(r,a) + m(r,a)
N(r,a) lim - - r+RT(r,f)
lim r+R
;;: T(r,f)
V(r,a) lim r-+RT(r,f)
+
It shows that we must have
V(r,a) lim - - r-rRT(r, f)
o (a)
=
if there are sufficiently many
o
a - points so that the expression
N (r, a) lim - - r-+ RT(r, f)
=
assumes the maximum possible value In the case f
n
R
= +00
1 .
we assume in the rest of this
§ that all
are non-constant. It follows then that T(r,f.)
~
J
o
N(r,f k ) lim r-+R T(r,f)
I) N( (0) , 1I N (00 )
In the same way we obtain for
(17.16)
and
N(r,a k ) - N(r,ak ) + 0(1)
N(r,a) - N(r,a)
0)
O(a)
:::
lim r+R
:;; lim r-+R
;;; lim
T(r,i) N(r,a k ) + 0(1)
r-+R ;;; lim
T(r,f)
Sea)
T(r,i)
T(r,f k )
r+R T(r,f)
N(r;a) - N(r,a) (ii)
fI N( 00 )
O(a), (a E C n )
;;; lim
T(r,f k )
( 17.17)
r+ R T (r , f. ) J
N(r,a) - N(r,a) ;;; lim
::: lim
T(r,f)
T(r,f.) J 153
S lim
N(r,a k ) ~ N(r,a k ) + O( 1)
N(r,a k ) + O( 1)
S lim
S
r->-R
T(r,n
J
T(r,L)
J
T(r,rk) lim r->-RT(r, L) J (17.18)
and N(r,f) ~ N(r,£) e( co )
= lim
(17,19)
T(r,£)
r->-R
T(r,n
In particular we see from (17.18) that 8(a)
j
O(a.,f.) J J
= 1, ... ,n
(17.20)
For the volume deficiency we obtain for
the estimates m(r,a k ) + N(r,a k ) lim ---,-'- - - - - r-~R T(r, f)
m(r,a) + N(r,a) (i)
?
lim r+R
T(r,n
= lim
T(r,f k )
T(r,fk) lim r->-RT(r, L) J
:-R T(r,n
m(r,a) + N(r,a) (ii)
lim r->-R
07.21)
m(r,a) + N(r,a)
lim
5
T(r,£)
T(r,L)
J
. m( r , ak ) + N ( r , ak ) ? 11m r-~R T(r,f.) J ,) (a)
For the deficiency
we obtain the following estimates for
m(r,a) (i)
o(a)
= lim
~
r+-RT(r,f)
m(r,a) (ii)
,) (a)
(17.22)
= lim
S
r->-RT(r ,£)
m(r,a k ) lim r->-RT(r,f)
m(r,a) lim r->- RT(r, L) J
-R T(r, f) r->-RT(r, f.) J (17.23) T(r,f k ) m(r,a k ) S lim lim r-T R T (r , f.) r->- RT(r, f.) ]
]
( 17.24) mer,£) ,) ( co )
= lim - - r+RT(r ,f)
154
N (r, f) I - lim - - r->-RT(r, f)
S
N(r,f.) I-lim J r+RT(r, f)
(17.25)
m(r,f) lim r+RT(r, £)
>
m(r,f k ) lim r+R T(r,O
(17.26)
In particular we see from (17.24) that 6(a)
(17.27)
6(a.,f.) J J
2
We have further for
a E
en
V(r,a) +N(r,a)
en
El(a)
N(r,a) -N(r,a)+ m(r,a)
=
= 1 - lim T(r,£)
r+R
lim T(r ,£)
r-~R
N(r,a) - N(r,a) + m(r,a) lim r->-R
~
;;;
lim r-+R
T(r,f.)
J
N(r ,a k ) - N(r,a k ) + mer ,ak ) + O( 1)
T(r,f k ) lim r-+ RT(r, f.)
T(r,f.) J
J
(17.28) N(r,a) -N(r,a) + m(r,a) El(a)
(ii)
= lim T(r,f)
r+R
;;; Jim r-+R
T(r ,f k ) lim r-+RT(r ,£)
N(r,a k ) + O( 1) + m(r,a k ) T(r ,£)
T(r,f k ) lim r+ RT(r, f.)
~
(17.29)
]
e( 00)
N(r,£) 1 - lim r-+ RT(r, f)
:;;
N(r,f.) 1 - lim J r+R T(r ,£) N(r,£) - N(r,£) + m(r,f)
N(r,f) e( 00
)
= 1 - lim - - r+ RT(r, f)
(17.30)
lim r+R
T(r,f)
N(r,f.) - N(r,f.) + m(r,f.) ;;; lim
r+R
J
J
J
(17.31)
T(r,f)
In particular we see from (17.28) that
155
e(a)
j :;: 1, ... ,n
e(a.,f.) J ]
(17.32)
These inequalities show thai the relative growth of the component functions f.(z) ]
and in particular the number T(r,fk ) min lim j,k r+RT(r,f.) J
(17.33)
has a very strong influence t on
o(a), e(aL
0v(a),
e(a),
0N(a),
(aECn~
from what we have just seen follows that these quantities must all vanish if (17.33) is zero. For meromorphic vector functions in the plane we could have deduced already from Proposition 14.15 that identically
provided that
p
Proposition 17. 9:
*
p
1:;
>
and 0
r;( z)
is a meromorphic function of order
, then
ar
160
, where
when
p
'0
ular a meromorphic function
tNevanlinna [27] ,po 51
r;
= 0 or I) (a , 1:;) :~ 1 - cos "TTP r; 1:; I:; is the only deficient value of r;( z) . In partic-
6(a,l:;) >0 I:;
P
r;( z)
of order zero can have at most one defi-
. Clent va1ue t .
We deduce Let £(z) = (f 1 (z), ... ,f (z» , (£1""'£ non-constant) n n 1 be a meromorphic vector function of order p in the plane, where 0;;; p < 2"'
Proposition 17.12
a = (aI' ... ,an) E en
Assume that for some 6 (a»
1 - cos
value of . ,n
f( z)
, and each
function
f( z)
P > 0
when
lTp
also,
• Then
6 (a)
a
>
0
when
p
= 0
or
is the only finite deficient
a. is the only deficient value of £.(z) for j = 1, .. J J has order p • In particular a meromorphic vector
£. (z) J of order zero can have at most one finite deficient point. 6(c)=0
Under the assumption we have
(c*a,oo) N(r,£)
cos TIP
+ lim - - - , (0 < p < .!.) r-+ +00 T (r , f) 2
+
N(r,£)
0 J J P = 0 or 6 (a.,L) ~ 1 - cos TIP when p > 0 . Theorem 17.ll shows J J that for each j=l, ... ,n the value a. is the only deficient point of J f.( z) . Thus inequality (17.27) shows that a = Car'" ,an) is the only J f( z) finite deficient point of ~
{a} ~ ~ ~ n (aI' ... ,an) E C
Now let
,~=1,2,
...
be the set of finite deficient points
for the vector valued meromorphic function
This set is either finite or is countably infinite. For each each
~
~
f( z)
j = 1, ... ,n
and
we have by (17.27) the inequality
~ Thus for each
t
a
6 (a: ,f.)
J
= 1, ... ,n
J
we have
Hayman [l6],p.ll4 161
L
o(a.R. ,£.) J
J
eEOC
0 (c,£.)
J
here the last sum is extended over all finite deficient points
c
of
We deduce the inequality
L
a(a)
min
.
L
f.( z)
J
O(e,£.) J
c E C
Using the estimates (17.20) and (17.32) instead of (17.27) we can do the analogue reasoning for the set of points set of points such that
EJ(a)
>
a
such that
O(a) > 0
,or for the
0 •
Summarizing and using Proposition 17.9 we formulate Proposition 17. 13
Let
be a meromorphic vector function in the plane. Then the following inequalities are valid.
L
a (a)
L a (c,£.)J
min
~
c EC
L
::;
e(a)
L
min
cEC
L
EJ(a)
~
min
e(e,f.) J
L
EJ(c,f.) J eEC
here the left sums can be positive only if all
have the same order.
Next we apply t
Theorem 17. 14:
Suppose that
in the plane, where
o
3 we have
tFuchs (23), Hayman [l6) ,po 90, Weitsman [59)
162
L6(a,r,;)(J, where
"3 we
f.( z)
A«(J"A.) - 6 (00,f.)(J, J J
+ - - 4 < wd[lwll ' wd[[w[[ >
[[wit
IIw[[
1 - - 2 < dw , dw > -
[[wit
dJlw['
-
d[[w[1 - - 3 < w"
dw>
[[wit
[lw[[3
(d [Iw[[) 2 +
[[w[[2 This gives for (8)
1
1
IIw[[Z
[[w[[4
- - < dw , dw > - - - < dw , w > < w , dw >
(10)
Remark.
The last expression could be abbreviated by
=
176
1 [[w[[4
Z
I[w
1\
dw[1
1
( 11)
~
From ihis calculation we conclude that we can define a Hermitian metric in pm-I
by the formula
==
~(LkWkWk)( J:kdw k 0
():kWk Wk)-2
dw k ) - (L k w k dw k
)0 (LkWkdW k ){ (12)
In fact, the right hand side shows that the metric is Hermitian and the left side shows that it is positive definite. This is the Fubini-Study metric of pm-I In terms of the left hand side of (12) the associated Kahler form can be written m-I 2
I C==I
m-I
'c..") C==I I
==
<poc!\ <Pac
<poc!\