Internat. J. Math. & Math. Sci.
Vol. 6 No. 4 (1983) 617-669
617
VALUE DISTRIBUTION AND THE LEMMA OF
THE LOGARITHMIC DERIVATIVE ON POLYDISCS
WILHELM STOLL
Department of Mathematics, University of Notre Dame
Notre Dame, Indiana 46556 U.S.A.
(Received June 18, 1983)
ABSTRACT.
Value distribution is developed on polydiscs with the special emphasis
that the value distribution function depend on a vector variable.
A Lemma of the
logarithmic derivative for meromorphic functions on polydiscs is derived.
Here the
Bergman boundary of the polyd+/-scs is approached along cones of any dimension and
exceptional sets for such an approach are defined.
KEY WORDS AND PHRASES.
Value distribution theory, valence function, Jensen formula,
characteristic function, counting, function, spherical image, compensation function,
Lemma
of the logarithmic derivative, and approach
cone.
1980 AS MATHEMATICS SUBJECT CLASSIFICATION CODES. 32H30, 32A22.
i.
INTRODUCTION.
Value distribution for polydisc exhaustions has been studied by Ronkin [i],
Stoll [2
and others.
They emphasized the growth of entire holomorphic and
meromorphic functions and the representation of canonical functions to a given
divisor in
Cn.
C. Ward Henson
For applications to mathematical logic, Lee A. Rubel and
[3], [4] inquired if the Lemma of
the logarithmic derivative could
be established for a meromorphic function on a fixed given polydisc.
In the classical one variable theory, the Lemma of the logarithmic derivative
has been one of the basic tools for a long time.
analogous Lemma was proved only recently.
In several variables, the
For ball exhaustions of
En,
Vitter
5
proved the Lemma for differential operators with constant coefficients and derived
the defect relation for meromorphic maps f
n
/
m
See also Stoll
6].
618
W. STOLL
Vitter’s Lemma extends easily to differential operators with polynomial or, with
proper modifications, to differential operators with entire coefficients.
Earlier, a weak version of the Lemma was proved by Gauthler and Hengartner [7] for
the special operator
D’
/z
z
I +
+
Zn /Zn,
extension to a general differential operator.
which does not permit the
D’
The differential operator
suffices
for meromorphlc functions, but imposes unnecessary restrictions for meromorphlc maps
f
cn
/p
m
Recently, Shlffman has shown how to derive Vitter’s Lemma from the
[8],
result of Griffiths and King
which also can be interpreted as a Lemma of the
Biancoflore and Stoll [9] gave an elementary proof of
logarithmic derivative.
The same method will be used to obtain the Lemma of the
Vitter’s result.
logarithmic derivative for polydiscs.
A theory should reflect the intrinsic algebraic, geometric and analytic
Our development of
structures of the mathematical landscape under consideration.
value distribution theory on polydiscs will adhere to this principle.
This is
an important feature of this paper, which is mostly self contained and requires only
a minimal knowledge of several complex variables.
Let us outline the main result.
by its coordinates
define
II
(Iz
Denote by
II
z
nl).
llrll
The euclidean space
the length of r in
/
Rn
partially ordered
For z
(z
z
I
n)
e
n define
For 0 < h e
RnO,)-- (r
n is
n
o <r
R
n
+
Let
n
be the rotation invarlant measure on the torus )<r> for r
/
has total volume I.
In
Rn(h),
introduce admissible approach cones M with
A M such that
then h can be approached from M
E.
_
we could approach h via th with t
However Rubel is interested in the more general approach r
On M we introduce a measure
n(h)
such that
only.
h with 0 < r < h.
We
< dim M < n which contain both cases.
AM(M)
.
If E
M with
AM(E)
<
,
619
VALUE DISTRIBUTION ON POLYDISCS
Let f be a non-constant meromorphic function on ])(b).
The characteristic
+
Tf(r,q)
of f is a function of two points r and q in
IRn(b).
q[l,n] and
Take T
+
8f/Sz.
Keep
the manifold M with
AM(E)
define
f
< a < b <
E with r >
such that for all r e M
< 17
log+Tf (r,)
+
19 log
we have
+
n
If S c_
{x
S(a,b]
{x
n,
n
E
define
If S
_
S
a < x <
b}
S(a,b)
(x
S
a < x <
b}
S[a,b)
{x
s
o}
s
+
(
s(0,)
S,
define x < y if
lq[l,n3.
S
x. -<
n,
3
x >
{0}.
If x
YJ
for all j
,xn) and y
(x
and if
%
a < x < b}
S
a < x < b}
s
x_>
(yl,...,yn)
l[l,n] and x < y if
(z l+w
and m
(Wl,...,Wn)
I
Zn+Wn)
{
s
in
n,
and p
6 lq
xj
<
yj
0}.
are vectors
for all
(IZl12 +
< }-
and
define
.m
(ZlW
P
(z,...,zP)n
Z
(%Zl,...,%Zn)
n
llZll
S
define
(Zl,...,z n)
z +m
E
(x
s[0,)
s()
For
If S c_
be the n-fold cartesian product.
+, define
S[a,b]
+
s
j
Then there is an open subset E of
fixed.
POLYDIS CS.
For any set S let S
in
<
log+,f/f,lln
<>
2.
n(b)
,ZnWn)
I)
+
IZn[2) I/2
eZ
(eZl,...,eZn)
620
W. STOLL
(log z
log
0
0
log
(0
n
log
Zn
Cn
0)
Here log is the principal value.
n,
(,)n
6
n
(I,...,)
n
For r
if
define the polydiscs
+
n
The Bergman-Shilov boundary is given by
A surjective, real analytic map
n.
all
If r > 0, then
r is
bijective on [0,2) n.
manifold such that
,
For r
D<r>
er n /)<r>
is defined by
r()
r-e
for
is an n-dimensional, oriented, real analytic
an orientation preserving, local diffeomorphism which is
D(1) with ID
we obtain the unit disc D
D[I] and 3
]D<I>.
+
For r
IRi, define the diagonal manifold
U
ID<tr>
teA(0, I)
A(r)
A surjective, real analytic map
8r(t,)
tr(#)
for all t
r
Air]
ID<tr>.
U
t [0, i]
n
IR(0,1) IR
IR(0,1) and
e
n.
A(r) is defined by
Moreover, A(r)
(n+ l)-dimensional, oriented, real analytic manifold such that
is an
8r
is an orientation
preserving local diffeomorphism which is bijective on IR(0,1) x [0,2)
On
(,)n,
n.
we introduce the holomorphic differential forms
dz
n
(2i)n
z
dz
A
A
z
n
n
(2. i)
VALUE DISTRIBUTION ON POLYDISCS
n (2i)n-I
lq[l,n]
For j
621
(2.2)
^
=I zu/
%=I
I
U
we have
dz.
(2.3)
Let
Jr
]D<r>
/
(,)n
Then
be the inclusion map.
* *
I
r
(2)n
dl
(2.4)
> 0
^ dn
A
(2.5)
D<r>
,
Hence
-n-ir()
is a positive measure on )<r> of total mass
invariant under the action of ]3<I
z e )<r>.
In particular, if F
n
>
)<r>
/
d
n
Let X be a form of bidegree (i,I) on an open subset U
0
X
x
X
0
zeU)<r>
]D<r>
0
is integrable, then
F(eiz)
n
Then
-
(z)
on 3<r> defined by
-n-ir()
z for
is
e )<i
n
>
and
(2.6)
of
cn where
n
i
is said to be non-negative
,
such that
,=I
X. dz
^ d
(2.7)
"
(respectively positive)
at
e U if for all
we have
n
U,=I
X(Z)xx
>- 0
(respectively > 0).
(2.8)
W. STOLL
622
If S E U and X is non-negatlve (respectively positive) at every point of S we say
that
X
is non-negatlve on S
(respectively X > 0 on S).
(respectively positive o__n S) in signs X > 0 on S
On U n
(,)n we have
(2.9)
/
Take r
t
n and assume that U
*.
rlr(X
A
dl
A
d
where
+
For 0
((I)
E
n
/
U be the inclusion map.
For
ei)rre i(-v) 2t dt ^ d
n
E
X(tr
U,=I
r(X
A
En
_>
0
if
X
lr( X
A
Kn
> 0
if
X > 0
on
{
/
u.
If
I
(2.10)
In particular
dn.
define h
{0}) c_ A(r).
A(r)
r
E
n (t,) (1)n
A
Let I
’n n we have
(I
](0, I) and
A(r).
by h
I(u)
> 0 on
A(r)
A(r).
EID<r>,
then
The identities (2.6) and (2.10) and Fubini’s theorem imply
easily
.
+
LEMMA 2.1.
Take r
neighborhood U of A(r).
E
Let X be a form of bidegree (i,I) on an open
Assume that
It( X
A
+
Take c
open neighborhood U of A(rr).
n and
is integrable over
A(r).
Then
(X) n()
^
COROLLARY 2.2.
n
(2.11)
+
r
IR.
Assume that
Let X be a form of bidegree (I,I) on an
Irr(X ^ n
is integrable over
A(rr)
Then
VALUE DISTRIBUTION ON POLYDISCS
x
.
’,
(rr)
Take n
l[l,p]
maps T
p
T
][l,p]
/
.[0,n].
and p
]
/
l[l,n].
0, then (p,n)
l[l,n].
Clearly T
T and
T
/
cp
T
/
.
n
p
-P,
,
then T
TO(U-m)
If
such that
z
Define
(6j I
j
O
is linear.
is called the support of
I,
.
n.
+
or by
Cp"
(2.12)
’6in)"
Cn-P
For each
an
is defined by
(2.13)
cP,
If % > 0 and n and m belong to
.[0,n] and map
z (j)
Obviously p
Im o
0 for j
0 iff
n Im o
Let F be a function on ])<r>.
Define
then
T0(n) I.
"[0(u 5k)
p and
Im O
Take r
For
n-p
T0(n) T0(m)
0 for j
n Im T
T
cn called the T-injection at
then one and only one number p
(j)
}.
called the T-projectlon is defined by
p
0
n
T__, w,() +
p=l
(1
If
n, then (n,n)
If p
(zT(1)’’’’’ZT(p))
the Kronecker symbol.
injectlve, affine map
[l,n], let
If p
(n-p, n) is uniquely defined by Im
T (z)
6jk be
.
Let (p,n) be the set of all injective increasing
If p
A surjective linear map
Let
(r)
various operators.
The complement T
Im T u Im
(x) ().
e<r>
lq[l,n] be the inclusion map.
(p,n), we assign
623
0.
(2.14)
e
(p,n)
n-p.
exist
The map
Also
(2.15)
624
W. STOLL
1
S(r F)
(2)
n
2 2
F(r e
il
r e
0
0
i#n
n
d
dn
(2.16)
provided the integral exists (perhaps in the sense of summability where +oo or
are permitted).
If r > 0, then
Fn.
(r,F)
<>
If 0 < r
#
0, then o
e
T(p,n) with p
e
(2.17)
El n]
Take 0 e
Cn-p
Abbreviate
Then
(r,F)
(F
m<
If r
() >
(2.18)
0, then
(0,F)
Take p e lq[ l,n] and T e
Hence F
o0)flP
TZ
is defined on
exists for almost all
z
T(p n).
)<nr(r)>.
D<(r)>
F(0).
(2.19)
Take z e m<z~(r)
>
then
By Fubini’s theorem
Tz0D<T(r) >) <r>
M(r(r)
F
and we have
M(r,F)
M((r),H).
(2.20)
More explicitly, this is written as
M(r F)
M(n(r)
M( (r)
F
Tz ))
(2.21)
If r > 0, then
m<r>
(2.22)
zm<(r)> m<r(r) >
Sometimes we shall write (r,F) as in (2.18) even if
r has some zero coordinates.
Then
(2.21) writes as
in
(2.22) which
is more instructive.
VALUE DISTRIBUTION ON POLYDISCS
3.
625
PLURISUBHARMONI C FUNCTIONS.
n.
Let B be a subset oflR
B +R u {_oo} is said to be increasing
A function g
+
if g(x) _<
g(y) whenever x
E
B and y e B with x
logarithmic convex if x e B and
B and
E
e
_<
.
The set B is said to be
(0,I) implies x
I l-I
e
B.
The
function g is said to be logarithmic convex, if B is logarithmic convex and if x
and
E
B and
B
(0,i) imply
e
g(xy 1-)
Xg(x) + (I- )g().
_<
(3.1)
+
n.
Here we are mostly interested in the case where B is a polydisc.
Then
n(b)
Take b e U
+
is logarithmic convex.
lq[l,n] and
Take p
T(p n)
T
Then
+
If 0
E
Cn-p,
an injective linear map T
inton(b).
T
Define
+
log
O
r(b) P.
cn
cP
is defined which maps
T
(b)
E
IRp
uP(T(b))
+
n(b)
Given a function g
+
define
h
LEMMA 3. I.
PROOF.
P(BT)
T
/m
u {_oo}
by
If g is logarithmic convex, then h
inP(BT).
Take x and
h (x
_<
In,
Observe, if T
then
g(T0(e x)
for all X
n(b)
+
If h
and y
Then
(e g) )
+
T0(u(T))--
Rn(b)
n(b)
with o
x
is the interior of
n(b)
q.e.d.
In
+
is convex for each T e
+
>’TO
%.
Define
%g(T0(eX)) + g(T0(eY)) XhT(x) + hT(Y)
particular g(e x) is convex and g is continuous on
LEMMA 3.2.
(3.2)
is convex.
Take % E(0,1).
g(T0(eXXeg))
+ g)
g(T0(eX))
hT(X)
o
y
+
Rn(b),
if g is logarithmic convex.
T(p,n) and p
and for all
q[1,n] then (3.1) holds
IR(0,1).
626
W. STOLL
z, x
the vectors
P()
T0(e x)
,
g(T 0(e xx+y))
g(x,y)
Define
Each function h
Xg(x) + g(g)
(0,0)%(I,i) I-%
g((0
q.e.d.
2 and
Take n
0 if 0 < x < 2
If 0 <
is constant and therefore convex.
T
exist in
hT(X + g)
0 and g(x,y)
0 or y
if x
coordinate of
Then
the conclusion of Lemma 3.2 may be incorrect:
and 0 < y < 2.
then
%.
Define
th
Hence x and
XhT() + hT() g(0(eX)) + g(0(eg))
<
(2,2).
0(e).
and
e XT 0(e g) )
g(r0(x)
g(xXy)
O
T where the j
Then
or y is positive if and only if j e Im T.
such that x
If
% i-%
x y
Define z
PROOF.
< I,
(0,0) and
0)%(I
i)
i-%)
> %
g(0,0)
%g(0,0) + (I
%)g(l,l).
However we have the following result:
LEMMA 3.3.
If g is increasing on
n(h)
and if h
+
g(p,n) and p
PROOF.
j
Im
and y
z
If
1
Define
z
e
and
xy
and z
j x.3
> 0 and
n(b)
yj
yj
with
o
> 0 if j
o
x
Im
z
y
o
x
Im
z and x
^
Im
z.
O
y
y.
for all
g(0) -< g().
0
(x I,...,
R
{_oo}
g(%)
Take b
+
n.
-<
yj
if
n n(h)+
Also we have
%g() + g() -< %g(x) + g(y)
q.e.d.
An u_er .s.emi-continuous function
increases and is logarithmic convex if and only if the function
+
u
and
x.
Put
Then x
x and y
P(8y)
Lemma 3.2 implies
PROPOSITION 3.4.
’-()n--
Then Im o
+
g(x%y )
g
is convex on
lq[l,n], then g is logarithmic convex.
I- 0, then (3.1) is trivial since g(0) -<
g()
(YI’’’" ’Yn
%.
T
re(h) +IR o {-} defined by u(z)
g(Izl)
is plurisubharmonic.
627
VALUE DISTRIBUTION ON POLYDISCS
If u is plurisubharmonic, a theorem of Vladimirov [i0] p. 88 (see also
PROOF.
Ronkin [i] Theorem 2.1.2) implies that g increases and that all
_
h%
are convex.
Hence
If g increases and is logarithmic convex,
g increases and is logarithmic convex.
then g increases and all
h%
By Vladimirov’s theorem u is
are convex.
plurisubharmonic, q.e.d.
Take h
+
n.
Let u be a plurisubharmonic function on )(h).
For r e
IRn(b)
define
+
Max{u(z)
M(r,u)
(2)
M(IZl;u)
Then
and
M(Izl,u)
IRn(b).
n(b).
M(r,u)
i)
(3 4)
dn.
d
-,
If u
then M(r,u) and
M(r,u)
are real numbers for all
If some coordinates of r are zero, the same remains true if u
PROPOSITION 3.5.
Then
u(re
0
are plurisubharmonic functions of z on )(b) (Ronkin [I]
+
+
r e
0
(3.3)
Therefore M(r,u) and M(r,u) increase and are logarithmic convex
pages 75 and 84).
functions of r in
n
)<r>}
2
2
1
M(r,u)
z
+
n.
Take b
Let u
)(b)
r0 _.
B be a pluriharmonic function.
n(b).
u(0) for all r
+
PROOF.
Since u and -u are plurisubharmonic we have
u(0)
M(r,u)
_<
A function g :(b)
u
plurisubharmonic functions u
-M(r,-u)
{-,+}
_<
-(-u(0))
u(0)
q.e.d.
is said to be quasipluriharmonic if there are
and v
such that g
u
v.
Strictly speaking
g is defined except for a pluripolar set
z g- {z
called the indeterminacy.
v(z)}
For 0 < r < h, the integral average
M(r,g)
is well defined.
u(z)
M(r,u)
M(r,v)
(3.5)
628
W. STOLL
Let S(b) be the real vector space of all quasipluriharmonic functions on (b).
Then the set p(b) of all pluriharmonic functions on (b) is a linear subspace of
S(h).
S(h)/p(h) be the quotient vector space and let 0
Let Q(b)
the residual map.
Q(b).
Take h
g
+
S(b).
0(g) with g
Then h
v where v
Q(b) be
An element of (b) is called a quasipluriharmonic class on )(b).
p(b).
n(b)
Take r
O(g) with g
If h
S(h) then
+
+
g
S(b)
n(b).
and q
M(r,g)
M(r,g) + v(0)
M(q,g)
M(q,g) + v(0).
Then
Subtract ion implies
M(r,g)
M(q,g)
M(q,g).
M(r,g)
Therefore the valence function of h is well defined independent of the choice of the
representative g by
N(h,r,q)
for 0 < r < b and 0 < q < b.
If r,
in h.
q,
M(r,g)
M(q,g)
For fixed r and q, the function N(h,r,q) is linear
+
p belong to
IRn(b),
then
N(h,r,q) + N(h,q,p)
N(h,r,q)
The set
S(b) and
S+(b)
S+(b)
Define
Q+(b)
If u
S+(b)
(3.6)
-
N(h,r,p)
(3.7)
-N(h,q,r).
(3.8)
of all plurisubharmonic functions u
on
)(b) is contained in
is closed under addition and multiplication with non-negative numbers.
0(S+(b))
in Q(b).
such that 0(u)
Each g
is called a plurisubharmonic class.
g and if 0 < r < b and 0 < q < b, then
N(g,r,q)
The definition extends to all q
n(b)
+
depend on the choice of u.
Q+(b)
M(q,u).
M(r,u)
with u
o
q,0
-,
(3.9)
and the extension does not
The function N(g,r,q) increases in r and decreases in q.
VALUE DISTRIBUTION ON POLYDISCS
629
If 0 _< q _< r < b, then
N(g,r,q)
_> 0.
(3.10)
If q is fixed and r converges to b, then N(g,r,q) measures the growth of g.
Now, we will give examples of valence functions of divisors, meromorphic maps
and meromorphic functions.
Take b
)(b).
+
n.
Let O(b) be the integral domain of all holomorphic functions on
Let K(b) be the field of meromorphic functions on 3(b).
field of quotients of O(b).
,
Define
{f
(b)
O(b)
f(z)
be the multiplicative group of units in O(b).
is written additively and is called the
K,(b)
D+(b)
D(b) be the residual map.
(@,(b))
is the
f
0.
is a homomorphism.
]D(b)}
0 for all z
Define
(3.11)
The quotient group D(b)
modul_______e o_f divisor
(f) for all f
f
K,(b),
If f
functions on re(b).
and let
O,(b)
D(b).
on
{0}.
O(b)
K,(b)/O
Therefore
,
If f
c
K,(b).
g/h where g
then f
loglf
loglg
(), then loglf
If
Df
Then
J().
We write
is non-negative,
0 and h
loglh
(h)
Let
additively closed subset of non-negative divisors.
the variable as an index,
we write
{0}
K(b)
K,(b)
Then K(b) is the
0 are holomorphic
S(b) and the map f +
Hence if
loglf
(h),
f
the valence function of the divisor
N(r,q)
N(p(loglfl),r,q
is well defined independent of the representative f
with
0 < r < h and 0 <
Ng(r,q)
v
f.
Obviously, if
< b, the definition (3.12) is also known as the Jensen formula
f
The map
(3.12)
re<r>
re<q>
is an additive homomorphism.
N(r,q)
+
N(q,p)
N(r,p)
n"
+
If r, p, q belong to
-N(p,r).
n(b),
then
(3.14)
W. STOLL
630
D+(b),
If
Nv(r,q)
then
r with
increases and is logarithmic convex in
N(r,q)
(3.15)
if 0 -< q -< r < b.
-> 0
Naturally, if we consider divisors on complex manifolds and spaces the
definition of divisors has to be localized.
+
Let V be a complex vector space of dimension m
Then
,
operates on
V,
by multiplication.
> i.
Define
V,/,
The quotient space (V)
{0}.
V
V,
is a
connected, compact complex manifold of dimension m called the compl.e.x projective
(A)
(A n
{IP(z)
V,)
0
vector functions V
w(b)
vector functions v
O,(b,v)
V
^
W- 0.
V,
The residual map IP
s_pace of V.
+z
V.
/
A}.
Define
and w e
Let O(b,V) be the set of all holomorphic
O,(b,v)
..representation of f.
Two holomorphic
are called equivalent v
O,(b,v)
from)(b)
{0}.
O(b,v)
O,(b,V).
This defines an equivalence relation on
is said to be a meromorphic map
If A c_ V, define
/(V) is holomorphic.
into
w,
if
An equivalence class f
(V) and each
V in f is said to be a
The representation v is said to be reduced if for each
representation w of f there exists a holomorphic function g such that w
meromorphic map has a reduced representation and if v and
gv.
are reduced
representations, there are holomorphic functions h and h on )(b) such that v
v
hr.
g
and h e O (b) has no zeros.
Hence hh
Therefore the non-negative divisor
the choice of the reduced representation
Let v and
.
Moreover w
w g
Each
gv
ghv.
hv and
Hence
is well defined independent of
Moreover w is reduced if md only if
be reduced representations of the meromorphic map f, then the
in det e rmin a cy
If
is well defined
[z
ID(b)
v(z)
analytic, with dim
f()
If
(v())
O}
{z
n- 2
(z)
]D(b)
If z
((z))
)(h)
IP(V)
O}
If
(3.16)
then
(3.17)
631
VALUE DISTRIBUTION ON POLYD%SCS
)(b)
is well defined and the map f
If
(V) is holomorphic.
+ O,
()
e )(h) with
representation of f and if
/
where
(V).
e
V,
The projective space (W)
O.
(V*) and if f()(h)
I, then (W)
Take a e (V*).
Then E[a]
If)
is said to be
ov depends on a and f only and
g
linearly
V,
then
is called the
The valence function of f for a is defined by
-
N
a(r’q)"
f
(3.18)
If w is a representation of f, there is a holomorphic function g
w
P()
(ker ) is a hyperplane in
E[a], then f
Nf(r,q;a)
Hence
is called a
Then a
If v is a reduced representation of f and if
The divisor
a-divisor of f.
m-
E[a] is a bijective parameterization of all hyperplanes in (V).
non-degenerate for a.
o v
V
is a linear map
The map a
If a
.
If p
Let V* be the dual vector space of V.
hyperplane.
IP(()).
then f()
Let W be a (p+ l)-dimensional linear subspace of V.
is a project.ive, plane of dimension p in IP(V).
If m is any
v.
By definition
w
g
a
Take a positive hermitian form (i)
V
V
/zlz ).
llzll
gv.
Hen ce
m + f"
(om
The associated norm is defined by
0 with w
(3.19)
,
called a hermitian product on V.
If x
V and z
V, we have the
Schwarz inequality
(3.20)
If
e V* and
B
V*, vectors
(z)
=
(zla)
e V and
B(z)
h
V exist uniquely such that
(zlh)
for all
e
V.
(3.21)
A dual hermitian product on V* is defined by
(IB)
(alb)
(3.22)
W. STOLL
6 32
I,
If
If z
E
then
I1=11.
1111
(V) and a
e
Therefore (3.20) and (3.21) imply the Schwarz inequality
V,
() with
(z) and a
(V*), then z
and
V,.
The
projective distance from z to E[a] is defined by
(3.24)
and e.
independent of the choice of the representatives
+
The compensation function of f for a is defined for r e
0 -<
mf(r,a)
M og
+Rn(b)
with
Oro(ID(Or(r)))
Then
logllvll
If.
is a plurisubharmonic
If v is another reduced representation of f a holomorphic function
h without zeros exists such that v
hr.
ogllil
log[h
(3.25)
log
Let v be a reduced representation of f.
where
by
[If,a]]
The definition extends to all r
function on lD(b).
n(h)
is pluriharmonic.
Hence
Then
ogli=ll
ogllli
/
and
log[hi
ogll=ll
define the same class
p(log]l I])=
The characteristic function of f is defined as the valence function of this class
Tf(r,q)
(3.26)
which is
Tf(r,q)
m(=,ogll=ll)
(3.27)
(3.28)
D<r>
D<q>
633
VALUE DISTRIBUTION ON POLYDISCS
The definition extends to all r e
n(b)
and q e
+
o
q0
(q)))
(m(
Tf(r,q)
+
Tf(r,q)
n(b),
_>
0
or0(3( r (r)))
O
If
and
then
(3.29)
-Tf(p,r)
Tf(r,p)
Tf(q,p)
Tf(r,q)
The function
with
+
If r, p, q belong to
If.
q
n(b)
if 0 < q _< r < b.
(3.30)
r and decreasing in q
is logarithmic convex and increasing in
and continuous where 0 < q _< r < b.
Let m be a representation of f.
holomorphic function g
$
Take a reduced representation v of f.
A
By definition
w
0 exists uniquely such that m
gv.
Dg"
Then
logllwll
)
]D<r>
C
n
ii II
lg"V’n
lglgiSn-
+
]D<r>
]D<r2
Therefore (3.28) and (3.13) imply
Tf(r
which generalizes
q)
’
re<r"
ogllwl
ogl[wll.n
n
N
<q>
W
(r,q)
(3.31)
(3.28).
Let v be a reduced representation of f.
linearly degenerate for a.
Tf_r(,q)
Take &
V,
with ()
\ ogllvlln
D<r>
Nf(r
q a)
’,
(V).
Take a
a.
$\
Assume that f is not
Take 0 < r < b and 0 < q < b.
og[
v[in
]D<q>
log[o vln
oglo
n
W. STOLL
634
.
mf(r, a)
mf(q, a)
log
log
so v
W<q>
Addition and subtraction imply the First Main Theorem
Tf(r,q)
Define T
b idegree
d
c
V / by
V
(I,I)
llzll 2.
TV(Z)
exists on (V) such that
(i/4)(- ).
on (V).
Nf(r,q,a)
Here
+
mf(r,a)
(3.32)
mf(q,a).
One and only one positive form
*()
dd
c
log
is the exterior form of the
Here d
V"
> 0 of
+
and
Fubini-Stud Kaehler
metric
We have
(3.33)
Denote the corresponding form on (V*) also by
.
Then
ram(a)
log
(3.34)
An exchange of integration implies
mf(r,a)m(a)
(3.35)
=i
Integration of the First Main Theorem yields
Nf(r,,a)n(a)
Tf(r,q)
(3.36)
a(V*)
For V
m+l
ZlW +
setting
write 1P
+
z w
m m
m
1P(V) and 1P
m
IP(V*).
On
m+i
Identify the compactified plane
define
(l:
{}
with IP
by
VALUE DISTRIBUTION ON POLYDISCS
z
m(z,w)
W
if z
and 0
+
w e
identify
?I with
u
,
setting
and a
,
(Zl,Z2)
(I 0)
If e e
-
{oo} by
2
IP()
If z e
.
0 and
z
(z,l)
z
(2)*,
define
re(w,0)
(I,0)
re(l,0)
(3.37)
and (0,I)
el
2
and
setting
i#
if
2
az
Then (e)
2.
i
IP()
and
0
then (z i)
(0 i)
635
with
(z,l)
a and
if
0
I
(2)*
Also
z.
e(z,l)
+
z
a.
so_-
is defined by
Define
oo
by
Then
Ic(z,1)
IIII ll<z,)Ii
./t
+
z
al
al 2
/Izl
(3.38)
2
+
(3.39)
l(z,)l
IIII ll(z,1)ll
ql + iz12
(3.40)
l%(z,o)
IIII
Hence the projective distance
sphere of diameter
in
3.
I
On
(3.41)
II(,o)II
I
on
91
is the chordale distance on the Riemann
the Fubini-Study form is given by
i
2
dz Adz
(i +
(3.42)
IZ12) 2
A meromorphic function f on D(b) is the quotient of two holomorphic functions g
and h
0
D(b) into IP
such that hf
g and v
(g,h)
which is identified with f.
D(b)
defines a meromorphic map from
All meromorphic maps from3(b) into
obtained this way except the map identical to m.
i
are
The representation v is reduced if
636
W. STOLL
and only if g and h are coprime at every point of ID(b).
a
f
n
re<q>
Jl lal2Jl
+
log
If
<r>
mf(r,)
log
<r>
_
P1
we have
(3.43)
f
g-ah
ID<r>
m e(r a)
For a e
Jl+ ]f12
lgJ Ig12 + lh12 n
Ifl 2
+
(3.45)
n
a
(3.44)
(3.46)
.n
Our definition of the value distribution functions corresponds to the AhlforsShimizu definition and is guided by the intrinsic nature of the algebraic and
geometric structures which are involved in the given mathematical situation.
The
classical definitions do not account properly of these features.
We would destroy
the intrinsic coherence of the theory if we would insist that q
0 and that
mf(r,)
be defined as the integral average of
log+Ifl.
Our definitions recognize the absence of a natural one parameter exhaustion of
the polydisc and take advantage of the fact that every holomorphic line bundle on a
If the polydisc is replaced by another complex manifold, the
polydisc is trivial.
definitions have to be localized and holomorphic line bundles become unavoidable,
also an appropriate exhaustion has to be chosen.
At present our definitions do not reflect the possible holomorphic slicings of a
polydisc and do not present the valence function as an integrated counting fur, ction
or the characteristic as an integrated spherical image.
These properties will be
studied in the next two sections.
4.
THE COUNTING FUNCTION.
+
Take b e
n and
a
(b).
Let g
0 be a holomorphic function on D(b).
there exists uniquely a non-negative integer
g(a)
Then
called the zero multiplicity of g
VALUE DISTRIBUTION ON POLYDISCS
at
=
such that for each integer j ->
g
637
(=) there is a homogeneous polynomial P. of
J
degree j such that
P. (z
g(z)
(4.1)
a)
J
j= (a)
g
for all z in a neighborhood of a in )(b) and such that
g
0 if and only if g(a)
(a)
0.
Ng+h(a)
Therefore the map
Moreover
Let
on
)(b).
+
@,(b)
gh(a)
g(a)
(4.2)
Min(g(a),h(a)).
(4.3)
g(a)
if and only if h(a)
Then v
is a valuation of the ring
@(h).
0.
f
0 is a meromorphic function
where f
Let g and h be holomorphic functions on )(h) which are coprime at every
point of )(b) such that hf
meromorphic map f.
u
Obviously
g(a) + Dh(a)
defined by g
be a divisor on ID(b).
0.
0 is a holomorphic function on B3(b) then
If h
gh(a)
P g ()
g.
Then (g,h) is a reduced representation of the
Let f, g and h be another choice.
@*(b) on )(b) without zeros exists such that f
reduced representation of f.
A holomorphic function
uf.
Then
is another
O*(b) exists such that (g,h)
Hence v
Cons equent ly
9(a)
(ug,h)
(a) -D(a)= g(a) -h(a)
(vug,vh).
.
(4.4)
is well defined independent of
the choice of f, g, and h and is
called the
.
multiplicity of v a__t a.
If
e
D(h) and p
(PVI
The map D(b)
defined by v
+
2
(=)
P)l(a)
+
2 (a)
The divisor v is non-negative if and only if (a)
v
(a) is a homomorphism:
7z, then
is the zero divisor if and only if
(a)
0(=)
_>
0 for all a
0 for all a e )(h).
identify a divisor v with its maltiplicity function a
way of defining divisors on complex manifolds.
(a) and
o.
)(h).
(4.5)
The divisor
Therefore we can
in fact this is one
W. STOLL
638
Let f be a meromorphic function on ]D(h).
a
f
is defined.
divisor
a
f()
Then
f
The function f
f(=)
f()
0 is holomorphic if and only if
-f().
0 and
fl
If
f2
a__t
a, the a-divisor
.
0, the
If f
a__t =.
is said to be the _multiplicity of f
holomorphic and without zeros if and only if
I/f()
If f
is called the a-multiplicity of f
is defined and
f
?I"
Take a e
> 0 for
(), and f
all
0 for all t e ID(I).
llf(i)
Also
0 are meromorphic functions on ]D() then
flf2 () flCa) + f2Ca)"
The map
K,(b)
defined by f
/
/
Let V be a divisor on )(b).
9(z)
+
f(a)
(4.6)
is a homomorphism.
0 is called the support of 9 and denoted by supp 9.
0, then the support of
)(b).
Let R(supp ) be the set of regular points of supp
.
a.
a
and c 6R
p.
divisor v on D(b) such that
)(b) such that
f.
If
f
supp
f
+
lq
of
is
If 9 > 0, then
If.
Take a e
I"
Then
O, then supp
Take p
9
Then the function
{z D(b)
() > 0}.
Let f be a meromorphic function on )(b) with indeterminacy
supp
If a
R(supp ).
0 then supp
If
pure dimension n
is an analytic subset of
constant on the connectivity components of
Assume that f
.
The closure in )(b) of the set of all z e D(b) with
If
supp
is
Of
u {Z e ID(b)
u supp
Let
(D(c))
Then f
f(z)
If
a}.
(4.7)
f.
)(c) /)(b) be a holomorphic map.
supp
9.
Take a
Let f be a meromorphic function on
0 is a meromorphic function on D(c).
The
pullback diviso r
q*())
(4.8)
foq
is well defined and independent of the choice of f.
pj
and
j
are divisors onD(b) with
(ID(c))
{
If
supp
9
V.3
>- 0, then *(v) >- 0.
for j
i, 2, then
If
639
VALUE DISTRIBUTION ON POLYDISCS
(m(=))
i supp(pl91
+
and
P22
+
*(Plgl
.
all u
that
_
En
For 0
+
If 0
()(t0(r)))
)(h).
said to be restrictable
,
Z
direction
For z
If u e
is defined.
and 0 -< t <
Rv(r)
)<r>
(t0(r))
t0(r)
with vertex 0 in
nv[z,t]
If
j
Let
Then
Rg(r)
is
Rg(r)
be
is a thin
abbreviate
v[,u]
[z](u).
define the counting function o__f
9 in
For
the
[z,u].
+z
+%
e
to).
ID(h)
C.
)(t0(r))
t
Here )<r>
and z
/
n[z,t]
There is a thin analytic cone C
n[0,t]
C
D<r>-
R(r),
n [%Z t]
n [Z
nv[,t]
Pll
+
P22
(0)
(4.11)
is a thin real analytic subset of
then %z
)<l%[r>- Rv([i[r)
l%It]
is restrictable to
and
<r>,
and
(4.12)
is non-negative and increases.
are divisors on ID(b) restrictable to z
I, 2, then
(4.10)
ue)[t]
n such that
0, the function
.
)<r>, then
is semi-continuous from the right and is of bounded
lim
0 <t->0
_>
exists such
supp w.
.
us for
R (r), the pullback divisor
variation on each compact subinterval of [0
If
>
(u)
as the finite sum
The function t
If 0
t0(r)
Take
(D(t0(r)))
)<r>
n[z,t]
for all 0
is defined by
be a divisor onD(h).
9
if and only if
real analytic subset of )<r>.
%()
En
and 0 < r < h, a largest number
Let
__t
+
)<r> such that w is not restrictable to
the set of all
V[]
z
an injective linear map
ID<r>
(4.9)
pl#*(l) + p2*(92).
P292
and if p. e
for
W. STOLL
640
n
Let f
z
pI vl+p22
[z t]
p n
i
0 be a meromorphic function on )(b) such that
<r>- Rv(r).
Then [z]
Df.
t0(r)
For 0 < q < r <
fo%
(4.13)
[z,t] + p 2 n [,t].
2
Take
the Jensen formula and
the definition of the valence function for [z] imply
loglf
Nv[](r,q)
)<q>
Nv[z](r, q)
The function z
zll
loglf
i
)<r>
is continuous on )<r>
Cv,
$
q
n[,t]
-
(4.14)
hence measurable on
Also
N[] (@r,r)
implies that
n[,r]
integrabie over
ID<r>
if r < @r <
counting function
n
n(r)
L,r]
of
If u
for
log
is integrable over
non-negative divisors
t0(r
Nv[](@r’r)
v
n[,r]
for
ID<>.
(4.15)
< @
on D(r).
is a measurable function of
n [z r] _<
Hence
r]
n[
log @
By (2.6)
N[](Or,r)
is
0 then
< @ <
t0(r)
(4.16)
r
Since every divisor is the difference of two
is integrable over
)(h) for all v
(h).
The
is defined by
nv(r)
\
0
zID<r>
n(tr)
$\
n[z
i] ().
n
(4 17)
n[z,t]n(Z).
(4.18)
Then
If 0 < q < r <
t0(r)
(2.6) implies
VALUE DISTRIBUTION ON POLYDISCS
(4.19)
q)n (z)
N[z](r
Nx)(rr,qr)
641
zD<r>
z)<r>
.
If p_ e
.
and
e
D(b) for j
n
If
_>
0, then
n(r)
P
r
q
q
t
i, 2, then
II+P22
> 0 and
r
Plnl) (r) +
(r)
n(tr)
(4.20)
P2nx)2(r).
increases in t.
+
PROPOSITION 4. I.
supp V n )<tb>
PROOF.
then [rb]
Let v be a divisor on (b) such that
O.
N(rb,O)
Hence
nErb,t]
Since 0
supp
_>
O.
If 0 < r < I,
By (4.17) we have
n(rb)
0 for
0 in (4.19) and
0 for all 0 -< r < I.
supp
connected neighborhood U of z
-< q and z
0
ZOI
such that r
< p < q such that qr < b and such that uz
0
lul
supp
we are permitted to take q
Assume that there exists z
0
with p -<
Then 0
0 for. 0 -< t < i.
,
.
Then v is the zero divisor.
for all t e IR[O,I).
W.l.o.g. we can assume that v
all 0 < r < i.
obtain
n.
Take b
supp
> 0.
ul
if p <
in )<r> exists such that uz
Then r < b.
-< q-
supp
Take
An open
for all u e
Rouche’s theorem implies
U.
n[z,t] n[z O,t]
for all z
Nv(qr,0)
U and t e [p,q].
> 0 by (4.19).
Hence n (tr) > 0 for p
Since 0
supp
q, which implies
A number s e (0,i) exists such that qr < sb.
0 <
which is a contradiction.
< t _<
Nv(qr,O)
Hence z
we conclude supp
supp
;
_<
N(sb,0)=
0
at most if a coordinate of z is zero.
q.e.d.
By the same procedures as in Lemma 10.4 and Lemma 10.5 of
result can be proved:
Then
2 ], the following
642
W. STOLL
PROPOSITION 4.2.
Then A
supp
A,
0
Let A
+n-I
is
0
A,
be the set of all
0 be a divisor on (b).
Let
I.
1 in 3().
Define
such that the restriction
0
Then A is either empty or a complex
locally biholomorphic.
manifold of dimension n
A
+
n().
is an analytic subset of pure dimension n
A- {0}.
A,
+
n
and r
Take b
The complement
0
A,
A
is analytic in
A,.
Also
n A(r) is either empty or an (n-l)-dimensional real analytic manifold which
can be oriented such that
n
0
defines a positive measure on A n A(r).
vn
+ 0(0).
be a divisor on (h).
Take p
nv(r)
We have
(4.21)
A0nA(r)
+
Rn and let
Take
p
n
p.
T )(T(h))
restrictable
(p,n).
Take
+)(h)
to
T(,n).
Then
is an injective
holomorphic map.
R(T)
If z
be the set of all z
3(())
3((h)).
We say that
Then
is
by T if
r(D(r()))
Let
Take any
1[ l,n) and define
R(T),
(())
supp V.
(4.22)
such that v is not restrictable to z by
i.
the pullback divisor
v[T,z]
Zz(V
(4.23)
exists.
PROPOSITION 4.3.
+
Take r e
n()
+
and
e
n().
Assume that
(r)
().
Then
Nv(r’q)
S
Nv[r,z](Uy(r) ,uT(q))~(z).
P
zD<p>
PROOF.
we have
A meromorphic function f
$
0 exists on /D(N) such that
(4.24)
f
v.
By (2.22)
VALUE DISTRIBUTION ON POLYDISCS
D<r>
D<q>
(q)
D< T (r)>
<gr(q) >
ze<p>
If
zID<p>
qT(r)
643
ze<p>
with 0 < q <
-
I, then (4.19) implies
I n[T,z]Ct(r))
N[,z](TCr)’qT(r))
dt
q
(b) define
For X
n T(p,x)
zD<p>
(4.25)
(4.26)
nvE T z](X)~.
P
We have
I
nv T(p tT(r))
q
Nv(r,q)
(r)
provided
If
0
(q)
()((b)))
p and
supp
,
T (q) qT(r)
T0(V)
then we have
VT
(4.27)
t
< q < I.
we are permitted to use q
n
If we abbreviate
with 0
d__t
0
(p t T(r)) d__t
t
(4.28)
N(p,0)
N(r,0)
N(r,q)_
which leads us
to a generalization of formula 4.2.3 by Ronkin [i]
N(p,0)
N(r,0)
Take p
is the T
st
and identify T
coordinate and
qT
{I}
(r
I
,rn) and q
(4.29)
t
lq[l n] with T(1) e qEl,n].
Then
i(r)
r
T
r
n
If r
d__t
nv T( p tT(r)
qrT" Hence
1
q
0
V ,T
(r
(p,tT(r))
-=
dt
rT_I,qT,rT+
nv,T (p’t)
,rn)
d_ t
then
(4.30)
644
W. STOLL
r
Ng(r,q)
q’twhere p
(r
dtt
ng,y(p,t)
(4.31)
,rn) and
,rT_l,rT+
nv, T(p,t)
nu[y,z] (t)n_
(4.32)
(z)
zD<p>
(4.33)
[T,z,u].
z](t)
n[%
uW[t]
If 0 < q < r < b are arbitrarily picked, we can represent
sum.
N(r,q)
as an integral
Define
Then r
0
r and
rn
q"
rT
(ql
’q%’rT+l
Pr
(ql
,q%_l, r%+l
(rr)
Also
rn
.(r T_I
p%
(4.34)
’rn)
(4.35)
Hence
r
N(rT_ rr)
n, r (pr
t) d-it
t
(4.36)
n,%(p%
t) d__t
(4.37)
By addit ion
r
Nv(r
5.
q)
%=i
q_
t
THE SPHERICAL IMAGE AND THE CHARACTERISTIC.
.
+
First let us consider the 1-dimensional case.
vector space of dimension m
Fubini-Study form on (V).
+
Take b
with a hermitian product.
Take a meromorphic map f
has dimension i, the map f is holomorphic.
Af(t)
f*()
ID(t)
Let
ID(b)
Let V be a complex
be the associated
(V).
Since
)(b)
The spherical image of f is defined by
V
t
[0,b).
(5.1)
645
VALUE DISTRIBUTION ON POLYDISCS
The function
Af
not constant and 0
<
t
Af(t)
Also
increases and is non-negative.
> 0 if and only if f is
If 0 _< q < r < b, the characteristic is given by
< b.
r
"t
q Af(t) t
Tf(r,q)
(5.2)
The identity (5.2) is easily derived from the definition (3.28) in applying Stokes
Theo rein twice.
+
n.
Take h
indeterminacy of f.
Let u
)(b)
i If,
then f
introduce
)(h) is
M
indeterminacy multipli_city
M, and v
If(=)
ID(h).
for each
a homogeneous vector polynomial
vo(z
v(z)
j=V
There is an integer
cn *of
uj
> 0 if and only if
= If.
Define
.
If(a)
Obviously
is an open, connected subset of
If G
is holomorphic such that
(5.3)
a)
]
for all z in an open neighborhood U of a in )(h).
D(h)
is a representation
0 and such that
degree j such that
G
If M is a
In order to counterbalance this effect, we have to
-> 0 and for each integer j ->
If()
be the
a holomorphic map such that
is a meromorphic function on
which may not be reduced.
If
V be a reduced representation.
connected complex manifold and if
(M)
Let
(V) be a meromorphic map.
)(h)
Let f
(G)
i If,
0 and
then
,
if
If(()).
o()
+
Take r
t__o
n(b)
if and only if
not restrictable to
and z
)<r>.
The meromorphic map f is said to be restrictable
()(t0()))
,
If.
The set
Rf(r)
of all
is a thin real analytic subset of
3<>.
)<> such that f is
For
3<>
Rf(),
a meromorphic map
f[]
is defined.
defined by
For 0
_< t
<
t0()
f
% D(t0(r))
re(V)
the spherical image of f in the direction
(5.4)
is
646
W. STOLL
Af[z,t]
+
Af[z](t)
f[]*() +
If(0)
If(0)
-> 0.
(5.5)
re(t)
+
If 0
D(t0(r))
ut
m<lul=>
then u
/
Af[z,t]
of f in the direction
/
= g
+
Tf[](r,q)
on
.
Uv[Z,u
Define the divisor
log
If 0
+
then
v[z,u]
v
0, we see that
)(t0(r))
define the characteristic
r
r
Af[z,t]
q
]
(u)
v[z,u]
then
_> 0
> 0
if 0
If(0)
> 0
0 we have
0 for almost all z e
>_ 0.
)<r>
For
D(t0(r)
on)(t0(r))
+
u e
(5.7)
Rf(r)
Let
the
be the
by
3(t0(r))
if u= O.
> 0 if and only if
> 0 if in (5.3) with a
u
[z]
z
vo z (0)
v
If(0)
0 is a representation of f[] on
vo
u
t0(r)
V be a reduced representation of f.
vector function v
divisor of
(5.6)
by
Tf[,r,q]
Let u :D(b)
For 0 < q < r <
increases.
and
Af[z, [u[t].
Af[uz,t]
The function t
Rf(lulr)
z(u)
vV(z)
0.
<r>- Rf(r).
uz e
If.
Since dim
If u
If
< n
0,
2 and
By (3.31) and (5.7)
we obtain
o II.o e,
Tf[z.r,q]
D<q>
])<r>
Since
re<r>
NpvEz
and
0 for almost all z in
o II= e, lln
]3<r>, the function
Tf[z,r,q]
(5.9)
is integrable over
(2.6), (3.27) and (5.9) imply
Tf(rr,qq)
Tf[z,r,q]n(Z).
zD<r>
(5.10)
VALUE DISTRIBUTION ON POLYDISCS
and if 0 < t < Ot <
Rf(r)
If z e ]D<r>-
Af[,t]
Therefore
Af[,t]
_<
Tf[,St,t]
t0(r),
then
(5.11)
I.
for 8
Af[,t]
log @
647
The spherical image of f is defined by
is integrable over )<r>.
(5.12)
Af[,l]n.
Af(r)
The identity (5.6) implies
for 0 < t <
Af[,t]n
Af(tr)
From (5.7), (5.10) and (5.13) we obtain
r
rf(rr,qr)
q
Af(tr)
+
If 0 < q < r <
and x
define
(5.13)
to(r).
dt
(5.14)
if x>_ r
log+_r_
L(x,r,q)
x
log+ q_x
log
PROPOSITION 5.1.
If r e
n(b)
f*()
^
A(r)
r
if 0 < x
q
t0(r)
(5.15)
-< q.
then
n + If(0)
(5.16)
’r’ql n If(0)
zA(r) llzll
"\
Tf(rr’qr)
If z
if q < x -< r
and 0 < q < r <
Af(r)
PROOF.
r
logx
f*()
L
llrl[
D<r>-
Rf(r),
then
f[]*()
(f
)*()
^
+
(f,())
lg
r
q
(5.17)
W. STOLL
648
Therefore Lemma 2.1 implies
(
Af(r)
If(0))n(Z)
f[z]*() +
zD<r> D
+
%z(f*())n (Z)
f
f*()
(0)
z<r>
A(t0(r)r)
If z
then
IZl
u(z)r with 0 < u(z) <
[II[
u()llr[l.
If 0 < t <
Define
X(x,t)
if q -< x -< t <- r and
t0(r)
f*()
q
A
^
r)
n + If(0))
ndTt- + Af(qr)
^
f*()
q
r
U(z) f*() ^
log
A(rr)-A(qr)
m(u(z),r,q)f*()
^
n + Af(qr)
n + A(qr)
n+
^
A(rr
log
f
log
log
_r f,()
q
r
q
^
n + If(0)
(0) log _r
and z
<r>-
Rf(r).
If
a
is restrictable to
Uf
mf[z] (r, a)
log
)<r>
is defined for 0 < r <
t0(r).
If 0
u
mf[uz] (r, a)
llf
with
<
’
3<>
log
IIf,all
n
_r
q
mf[z](lulr,a).
log
Take
then
t0(r),
Also (2.6) implies
mf(r,a)
z,
(5.8)
i
z’ all
lu[t
log
q.e.d.
q
(V*) and assume that f is not linearly degenerate for a.
+
(n(b)
Then
A(rr)-A(qr)
/\(rr) -A (qr)
r
Moreover
0 if q -< t < x -< r.
X(x,t)
X(u(z),t)
Take a
t0(r).
A(tr) if and only if 0 < u() < t.
then
X(u(z) t)f*()
q
n + If(0).
^
A(r)
then
(5.19)
649
VALUE DISTRIBUTION ON POLYDISCS
Now,
mf(r,a)
mf[](l,a)n()
(5.2o)
mf (rr,a)
mf[] (r,a) n()
(5.21)
z<r>
we will slice the characteristic
Here
parallel to the coordinate planes.
we encounter the difficulty, that the pullback of a reduced representation of a
We will show that the
meromorphic map may not be a reduced representation anymore.
problem does not occur for "almost all slices".
Some preparations are needed.
I.
For the technical apparatus we refer to Andreotti-Stoll [i
+
n+
Take h
LEMMA 5.2.
and r e
n(h).
Let A be an almost thin subset of 3(h).
Then A n ]I)<r> has measure zero on ]D<r>.
By definition there is a sequence
PROOF.
0
that A
A
of thin subsets of )(h)
such
there exists an open neighborhood U (p) of p in
A
For each p
{Av}eq
=i
)(h)
and analytic subset
dim B
A
B(p)
Hence B (p)
(p) < n.
of
U(p)
such that A
3<r> has measure zero on
3<r> has measure zero on 3<r>.
n U (p)
Hence A
U (p) covers A
B(p)
n U (p)
<r>.
and such that
Consequently,
A countable union of these neighborhoods
n 3<r> has measure zero on 3<r> for each
q.
)<r> has measure zero on )<r>; q.e.d.
Therefore A
+
n.
Take h
IR
Let f
3(h) /(V) be a meromorphic map.
reduced representation of f.
(p,n).
For
such that
Then
e
3((b))
(,n).
Take p
lq[l,n).
Then p
n
Let v
p
3(h)
[l,n).
V be a
Take
We have the surjective projection
~
D(h)
/m((h)).
(5.22)
we have an injective holomorphic map
)(T(h))
Zm 3(T(h))
ID(h)
(5.23)
I()
(5.24)
W. STOLL
650
Define
is biholomorphic.
Then
Rf(T)
c_
Rf(Y)
{Z
])({(b))
Tl(If
Sf()
{z
])(())
dim
Sf(T).
Rf(T),
)(())
If
]D(T(b)) }
Tl(If)
(5.25)
I}.
> p
(5.26)
the meroorphic map f restricts to a
meromo rphic map
f[y,]
If
6
Sf(),
D((b))
reduced representation
the reduced representation
)(b)
oz ID(())
By (5.24) we have
{z
Sf(T)
LEMMA 5.3.
PROOF.
Sf(T)
x
6
If
n
e
nl(z)
dim
is almost thin in
E’
V of f[T,].
/
)((b))
By [I] Theorem 1.14 m
and by [I Lemma 1.30
(E)
at
x.
If
{
N c_ X. for some j.
{l(z)
I.
_> p
n U
X
<
Sf(T).
is analytic in
If
Then
An open, connected
u
r and where
-
I}
u
X.
Xr where
each
3X"
is
is locally irreducible
n U exists with x e N and dim N > p
x
_< rank
< dim
E and z
(x)
7~
T
IX.3
Take r
dim X
j
If
dim N < n
E’.
Hence
+
THEOREM 5.4.
If
zllf
x
I}.
I.
Then
By [i3 Lemma i. 7 we obtain
rankx II
’ f
Therefore x
> p
Take z e
l(z)
n
I,
n U with x e X. for j
If
rank
If
If
x
l(z)
0
is almost thin.
exists such that dim
A branch N of
If
V restricts to a
])((b)).
neighborhood U of x inD(b) exists such that
a branch of
(5.27)
+(V).
T )(nT())
f
IRn(b)
Sf(T)
dlmx Xj
2
.I()
(p- I)
{-
I.
c_ E’ is almost thin; q.e.d.
+
and q
Tf(r,q)
zD<p>
n(b).
Assume that p
Tf[r,z](y(r),y(q))~(z).
P
_(r)
_(q). Then
(5.28)
VALUE DISTRIBUTION ON POLYDISCS
PROOF.
z
m<p>
ID()
Let
Sf(I).
Then
651
V be a reduced representation of f.
f[I,z]
exists and
v
Iz
Take
is a reduced representation of f.
Therefore
p
Now2.221
implze
zogll,,ll Tf(r,)
D<p>
D<r>
z()
If
p
o_i(r)
with 0 < q <
q.e.d.
D<q>
I, then (5.14) implies
Tf[i,](%(r) qT(r))
d__t
Af[% ](t%())
q
t
(5.29)
Define
Af, %(p,t%(r))
Af[,](tT(r))
d__t
.t
(5.3o)
Then we have
dt
rf(,)
provided
nT(r)t
_()t
Take p
Af,(,t[()) T
q
and z ()
(l,n)
and identify
with 0 < q <
qua(r)
%
with
(5.3,)
(I)
IN[l,n].
e
Then
r
Af, Y (
If r
(r
n)
r
and
twy(r))
(r
q’r
ri_l,q%,r%+
Tf(r,8)
where
(r
,r%_l,rT+l,...,rn)
:
Af,
"[
( t) d-it
t
,rn)
(5.32)
then
r
q’r
Af,i(,t)
d_ t
(5.33)
and
dt
Af% (;, t)
Af[%, ](t) T
z<W<p>
(5.34)
W. STOLL
652
(5.35)
fET,z]*(m).
AfEt,z](t)
DEt]
Take 0 < q < r < b and define
rT
and
r
Af,(pT,t)
Tf(r,q)
Take a
p e IN[l,n).
by
n
Then p
is not linearly
p c IN[l,n).
Take
dt
(5.36)
Take
degenerate for a.
T(p,n).
c
We obtain
Then
%
T(p,n).
e
Take
+
+
b
(V*) and assume that f
-
as in (4.34) and (4.35).
p%
R
and r
.
n(b).
Take z
D(~(b))_
%
Sf(%)_
such that
z
is restrictable to
Then
mf[r, ] (,a)
log
Iif z, all
l:)<p>
(5.37)
P
+
P(r(b)).
is defined for all p
Then (2.22) implies
mf(r,a)
z<(r) >
6.
(5.38)
mf[r,z](%(r),a)(z).
THE LEMMA OF THE LOGARITHMIC DERIVATIVE.
Take 0 < r < R < b <_ +.
Let f
$
0 be a meromorphic function on 3(b).
By [9]
Lemma 2.1 (see also Hayman [I, Lemma 2.3) we have
log+If /fla
<
log+mf(R,0) + log+mf(R, )
+
3
log+nf (R, 0)
(6.1)
3<r>
+
3
log+nf(R, )
+ log+
R
(R- r) 2
+
lg+
+
lg+
r
R
r
+
8 log 2.
+
We keep the notation of the last three sections.
n
Take b
and
and
in
+
n(b).
Take
IN[l,n] and
0 < r
r (r) <
assume that
(R)_
R < b
~(r)
T
p
T(R).
(6.2)
VALUE DISTRIBUTION ON POLYDISCS
0 be a meromorphic function on ID(b).
Let f
653
Abbreviate
0
f
THEOREM 6. i.
log+If/fl%
f
f
f
(6.3)
z
T
Under these assumptions we have
-< io
g+mf (R,0)
+
io
+ 3 log+
g+mf (R,)
n,
T
(6.4)
(p R)
D<r>
+
For z e
PROOF.
f[i,z],
f
3
log+n
<p>-
,
(p R) + log+
Rf(r)
r)2
fiT,z]
we have
+ log+
R
(R_
io g
+
r
-r +
16 log 2
The chain rule implies
Tz
f
+
Now (2.22), (4 26) and (5 38) imply
T
log+l fr/f In
)<r>
D<r>
D<p>
+
-<
mf[y, z] (R, 0)n_ +
log
+ 3
log
+
+
re<p>
+
(R’)
+
)<p>
log+mf (R, 0)
+ log+
+
R
(R- r) 2
THEOREM 6.2.
g+mf (R,)
+
log+
log
+
i
r
io
+
+
+
log
3
+ log+
r
R- r
3 log
R
2
(R- r)
ng,T (p,R)
r
2
n[y,z](R)n_
re<p>
+
+ log+ R-r
+ 8 log
+
re<p>
3 log
ng[,z](R)n_
re<p>
mf[y, z] (R’)n- +
+ log+
n[T,z](R)n
io
+
n-
(R- r)
mf[r, z] (R’0)n-I +
3 log
R
log+
+
n[y,z](R)n_
re<p>
log
g+mf[ y, z
re<p>
re<p>
_<
io
+
lg+--r +
+ 3
io g
+
log
n, Y (p
+
r
R- r
+
16 log 2
R)
+ 16 log 2
q.e.d.
If 0 < q _< r, we obtain under the same assumptions
log+IfT/fln
_<
8 io
g+T f (R
+9
log_
q) + 4 i o
g+mf (q 0)
+ 4
log+mf( )
D<r>
2R
r
+ 2 log+
7+
24 log 2.
(6.5)
654
W. STOLL
Define @= (R
PROOF.
+ r)/2R < I.
Then r < @R < R and OR
(R
r
r)/2.
Define
,rT_I,OR, rT+
(r
Then r <
Since
<
Nv(,)
>
()
r <
(r)
and
--OR <
C)
rn).
-
0, the First Main Theorem implies
mf(,0) < Tf(,)
+
mf(,0)
Tf(,)
<
~C)
R and ~(r)
+
mf(,,O)
log+mf(,0) log+Tf(,) + log+mf(,0)
+ log
log+mf(;, ) log+Tf(,) + log+mf(, )
+ log 2.
<
<
~()
(6.6)
2
(6.7)
By (4.31) we have
N(R,q)
Ng(R,r)
>
+
N(r,q)
R-@R
R
n,T (g’@R)
(,OR) -<
io
Ng(R,r)
_>
g+N (, )
lg+
2R
R
--_>
@R
n)
y(t, t)
d-it
t
R- r
2R
n ,’[ (
+
n9 T(p,t)
r
r
-< io
g+(Tf (R,q)
+
mf(q,0))
2R
+ log Rr
or
log+n
1o
(p,OR)
g+n
_<
log+Tf (R,q) + log+mf(q,0)
(p,OR) -< io
g+Tf (,) + log+mf(q, )
Also we have
log
OR
+
(OR
r)
2
<
OR
log+ OR-
+ R +
log
r
r
+
R- r
+
log
Now, Theorem 6.1 for r and
and
R
+ log+ OR-
log+
r
R- r
2R
R- r
<_
+ log
2R
(6.8)
2R
+ log R+ log 2
r
(6.9)
r
+ log+ R
+
log+
r
(6. I0)
2R
<- 2 log
+ log+ r
R- r
2R
(6. II)
log R- r
(6.6)-(6.11)
imply
+ log
2
R-
(6.5),
q.e.d.
VALUE DISTRIBUTION ON POLYDISCS
and 0 < q
< 0
Take
n.
Take h
THEOREM 6.3.
n(h)
Take r
Take
log+IfT/fln -< 8 log+Zf(Or,)
+ 4
with 0 <
-< r < @r < h.
q[l,n].
%
n(h)
and q
R such that 0 < q" h <
meromorphic function on )(h).
655
Let f
8f
f%-----.
Define
log+mf(,0)
+ 4
< r < h.
0 be a
Then
log+mf(, )
(6.12)
<r>
+ 9 log
PROOF.
20
O-L I +
+ --+ 24 log 2.
2 log
Theorem 6.2 can be applied with r
rT
qb T
and R
Or
T
and
R-- _(r l,...,rr_ I,R,rT+ l,...,rn )"
Then
R
<
Or and
Tf(R,)
<_
Tf(Or,).
I) and I/r
20/(0
Also 2R/(R- r)
_<
I/qbT.
Consequently (6.5) implies (6.12); q.e.d.
Theorem 6.3 is a preliminary version of the Lemma of the logarithmic derivative.
If we restrict ourself to the approach
For r to approach h, we have to eliminate 0.
rb
b with
> r
1 this could be accomplished along classical lines and would
Rubel and Henson want to consider the
result in the usual "exceptional intervals".
approach h > r
In order to satisfy both cases, we introduce a more general
h.
method resulting in "exceptional
Take h
+n
R
and p
.[0,n).
sets" of
and n.
dimensions between
Define 6
Rn(h) +(0,I]
by
+
Min{1-
6(x)
for x e
n(h).
(xj/bj)
J
l,...,n}
(6.13)
k
Let B be a real, pure p-dimensional, oriented submanifold of class C
+
in
n with k
I.
Denote by
B
the differential geometric measure on B.
the p-dimensional Hausdorff measure.
N
the normal space of B at
n Tx Nx.
Let _O
x
.
For
e
B, let T
It equals
be the real tangent space and
Then we have the orthogonal decomposition
n Nx be the projection.
The manifold B is said to be an
approach base if the following conditions are satisfied.
_
W. STOLL
656
(aY
The manifold B has finite volume
(b)
If x
B, then 0
_<
x < b, that is, B
(c)
If
(d)
If p > 0, there is a number
(e)
If p > 0, then
B and t
+ t(h
and
<
B(B)
n(h).
+
) e B, then t
0.
> 0 such that for each
log(I/6)
e B we have
< oo.
dB
(6.15)
B
The closure B is compact and contained in
n[h].
If
e
h + (
B, then b
N)
B
+
which contradicts (c).
Hence
Define the associated approach cone by
M
If
{ + t(h
B, then x < h and
EXAMPLE I. Take
)
+ t(h
<
B and t
(0,i)}.
(6.16)
+
) <
for 0 < t < I.
Hence M
_n(h).
n
n(b)I
B
+
j=l
xj/bj
n
M
n(h)
X
I
+
x./b. >
j=l
J
3
Simple calculations show that the conditions (a)-(e) are satisfied.
approach base and M the associated approach cone.
Observe that M u
Hence B is an
{h}
is an open
+
neighborhood of h in
EXAMPLE 2.
IRn(h)
Take B
u
{h}.
{0}.
Then M
{tb
0 < t < I}.
While the geometric meaning of the conditions (a)-(d) is obvious, the
condition (e) is more difficult to analyze.
For this purpose we offer the following
cons iderat ions
LEMMA 6.4.
Let B be a real, pure p-dimensional, oriented submanifold of class
C
k
in
-
-> I, with p
with k
_>
B()
< m.
B(B)
and with
ran((1
B
For each %
(0,i] define
)b).
+
Assume there are constants c > 0 and q > 0 such that
%
657
VALUE DISTRIBbION ON POLYDISCS
(0, I].
B(B(%))
_<
c%q for all
Then (6.15) is satisfied.
PROOF.
Observe that B(1)
A(m)
d
B
m
(B(1/m))
B(A(m))
a
m
{x
B(i/(m + I))
B(I/m)
{x
B and B()
cm
<_
%}.
q we have
For m
I/(m + I) < (x)
B
e
(x)
B
<
l/m}
-q
d
m
dm+ I"
This yields the estimate
log(i/) dn B <B
%
log(m + I) dD B
m=
a
m
m
A
m
lim
r-o
( rm_
d
m
m
log(m + I)
log(l + i/m)
m
dr
+ l/m) <-
N x
g((0,1)
definition M
B).
g(,).
x)
Then t(b
t
t
t and
x
+-
(b
x)
X
x; q.e.d.
q .e.d.
-
Nn
x
x +
+----It (
(b
x +
x) for t e
and x
then x < (t,x) < b.
If 0 < s < t <
b.
(
+
).
(b
t
and
t
By
(b
x))
R such that
Hence
)
B.
n.
IR
is injective.
Assume that x and x exist in B and
PROOF.
-l-q <
x + t(h
by y(t,x)
(- {i})
The map y
LEMMA 6.5.
Hence t
x
I))
m=l
/ is defined
x and (l,x)
Obviously (0,x)
X
n
log(r +
)
m
c
m=l
A map y
log(m + I)
din+
m=
-q log(l
_< c
,
r-I
1
lim(m
-r
d
g(t,x)
log(m + I)
W. STOLL
658
LEMMA 6.6.
PROOF.
IR(0,1)
shown that
U(0,1)
The map
M is a homeomorphism.
B
By definition the map is surjective, hence by Lemma 6.5 the map
B
Trivially the map is continuous.
M is bijective.
Let U be an open subset of(0,1)
is open.
B.
U such that
B
subsequence we can assume that
(t,x)
e
[0, I]
U.
B
Then
t
(t,)
and x
for
/
(t,).
(t,)
(,)
and
Therefore
which is wrong.
(tg,)
for
.
v
e U.
{(t9,9)}961q
By going to a
with
(t,x)
(I,)
I, then b
If t
(t,x)
(t,x)
A sequence
Assume that (t,x) is not an interior point of (U) in M.
exists inlR(0,1)
Take
It remains to be
6M
e U which is impossible.
Therefore (’U) is open in M; q.e.d.
Let V
and let g
gv.
+@
be an open connected subset of
V
t
(t,V)
an open subset of B
gv
+n hy w(t,v)
1
Let
(t,g(v)) for
Then
v)
g(v)
b
^Wvl(t,) ^
The vectors
IR x V
Define w
be the partial derivatives of g.
wt(t
W
@ be
Let U
U be an orientation preserving diffeomorphism of class C
and v e V.
t
P.
Am
Vp
gv
(v),
(t v)
w
V.
(t,)
t)P(b
(I
(v) span
By (d)
Tg(v
p
g(v))
(I
^ gv
t)g
(V)
^
v
(v).
(6.17)
^ gvp (V).
(6.18)
g(v)
we have b
T
g(v)"
Therefore
(b
In fact if x
g(v))
A
gVl (v)
A
A
gv
+
(6.19)
0.
g(v), then
I (A(x)
g(v))
IIgv
^ gv
(v)
^
(v)
^
^ gv p (v)II
(6.20)
^ gv (v)II
p
depends only on x and not on the parameterization g.
defined.
(V)
P
Hence A
B
is
globally
VALUE DISTRIBUTION ON POLYDISCS
LEMMA 6.7.
PROOF.
We have b
(6.21)
.
P
x
0(b_
) +
a
j
j=l
II(or
^ gv
x)
IIo(
A()
(v)
^ gvp (")ll
^
)If.
<-Ii11,
(u)
Hence
j
ll0(
_>
x)
llgv
y.
Also
n
)112<- Ii - 112
A(X)
gv
By (6.14) we have A(x)
A()2
which implies
659
If x e B, then
(b
xj)
(.)
^ gvp (")II
^
n
2
_<
b
2
2
q.e.d.
By (6.18) and (6.19) the map
IR(O,I)
x B
_n
is smooth.
In conjunction with
Lemma 6.6 we have proved:
THEOREM 6.8.
dimension p
+
M is an embedded, oriented, differentiable manifold of pure
and of class C
(0,I)
such that
B
M is an orientation
preserving diffeomorphism of class C
Let
M
and
the differential geometric measures on M and B respectively.
B be
In the situation (6.17)-(6.20) we have
dB
Ilgv ^
dM
(i
t)P[/(
^ gvp I
g)
dv
^
^ Sv ^
A
dVp
^ gv p II
dt
^
dv
^
^
dv
P
Hence globally
dDM
(I
t)PA()
dt
UM (M)
p+
^
(6.22)
dB.
Now, Lemma 6.7 implies
0-<-p
Here we are able
condition (e)
B (B)
<
UB(B)
<
.
to give another example where the
conditions
(6.23)
(a)-(d) imply
660
W. STOLL
+
PROPOSITION 6.9.
Take
submanifold of class C
k
c
n.
Let A be a pure (n-l)-dimensional, oriented
n with k
in
_>
Assume that B
1.
A n
n(h)
is relative
+
< oo.
B(B)
compact in A.
Then
(6.14) holds.
Then (6.15) is satisfied.
> 0 such that
Assume that there exists a number
If also condition (c) holds, then B is an
approach base.
PROOF.
B
B
u
Define
Bo
{
c B
u B
Each
B.
is a compact subset of A.
n
1- (x./b.)} for j
()
Then
n.
B
Take a
Then there exists an orientation preserving diffeomorphism
with
J
a.
b
3
J
U of class C
V
where U is an open neighborhood of a in A and where V is an open neighborhood
n-I and
where g(0)
=.
of 0 in
G
llv
G.1
(-i)
Let V be the gradient.
A
i-I
^
Bvn-i I
det(Vg
Abbreviate
> 0
Vgn)
Vgi_ Vgi+
Then (6.20) reads
n
Assume that
Vgj(O)
continuity.
Also
Since A() _>
0.
Vgj(0)
0 implies
0 < A(a)G(0)
(bj
This contradiction shows that
Consequently log(b
then
a.3
compact,
<
b.3
and
log(bo
3
gi)Gi I"
! (bi
i=
g)G
(A
> 0 for all
G.(0)I
0 for all i
gj(0))Gj(O)
Vgj(O) #
B,
aj)G.(O)
(bj
x.) is locally integrable at
J
B
3
log(bj
xj)
is
J
x.)
0.
0 at a.
3
if a.
3
b
trivially locally integrable at m.
x.) is integrable over B..
3
> 0 by
_>
Therefore
j.
Therefore d(b.
J
O.
A()
we have
J
If a
Since
J
Bj
b
j’
is
Therefore
3
log(i/6) d -<
B
B
log
j=l
B
3
b.
Xo
J
dB <
J
If E is a measurable subset of M, then the set
E
{t
m(0,1)
+ t(h- x)
E}
q.e.d.
VALUE DISTRIBUTION ON POLYDISCS
x e B.
is measurable for almost all
661
E /IR be integrable over E.
Let g
Fubini’s
theorem implies
p
g(y(t,x))(l- t)
E
E(x)
xeB
dt)A(x) dB(X
(6.24)
).
Define
d(r)
(E)
Then 0 -<
(E)
<_
LEMMA 6.10.
Define
Min{llh
0
2
t
}-
e
(6.25)
p+l
Ilb-
M[r,s]
Then
Y2
AM(E)
d()
> O.
Define
YIIhlI-P-I"
YI
llh- II
()
Y
]lhllp+l
Y0
Then
Recall that 0 < y -< A() <
For 0 < r _< s <-
-ii h
xllp+l
(6.26)
t
-<
llhll
-<
llp
ii h
E()
Hence
<(0)
p
YI
(6.26); q.e.d.
define
y([r,s]
x
B)
y((r,s] x B)
M(r,s]
M(O,I) and M o {h}
M(0,1].
oo, then
x
x B
which with (6.25) implies
j
xll
E()
2
Then M
m(x)
(I/Yo)P
Let E be a measurable set on M.
PROOF.
xB
oo.
Y1
B
I- t
rllP+l
lib
rE
{M(tj,l]}jq
If t.
M(r,s)
y(IR(r,s) x B)
M[r,s)
([r,s)
IR(O,I) for j
B).
and if t.
is a base of open neighborhoods of h in M u
{h}.
for
If
0 < r-< s < i, then (6.26) implies
Y2B(B) log
54(M[r,1))
r _<
s
AM(M[r,s])
_<
YIB(B) log
(M(O,s])
<
.
r
s
(6.27)
(6.28)
W. STOLL
662
LEMMA 6. II.
Define
Then
<
(Q)
.
If
e
6()
B, if
(6.29)
B and 0 < t < I- 6()}.
{(t,)
Q
(t,), then r
and if r
<
< t
e M- Q and
(6.30)
According to (6.26) we have
PROOF.
dt
t
dB(
log(I/6) d B <
i,
B
+ t(b
Also r
(I- t)(b- K) where (I- t) <- 6().
h- r
6() < t < I.
M- Q if and only if 0 _<
)
If so, then
Hence
q.e.d.
Let g
We say that g(r) -< h(r) for most r in
M / be functions.
M / and h
M and write g(r) < h(r) on M if there exists a measurable subset E of M with
such that g(r) _< h(r) for all r
6
If g -< h and h -< k on M, then g <- k on M.
(b)
<
k on M, then gk -< hk on M.
If g < h and 0
(c)
If g <- h and k <-
(d)
If t
(e)
If
g(r)
(f)
If
g(r)
on M, then g
(0,I) and g(r)
for r
<
h(r)
/
+ k -<
b with r
{tj}je
with 0 <
exists.
Then r.
t.3
<
such that
b for j
lim inf
Mr/b
g(r)
then g(r) -< h(r).
and 0 < g e
then a
and
_< lim inf
j-+oo
<
6g(r).
Mr+b
In the case (f), there exists a measurable
such that g(r) < h(r) for all r e M- E.
<
AM(E)
M[t0,1),
lira sup h(r)
g(r)
(a) to (e) are trivial.
set E in M with
e
M and if a e
Mr+b
PROOF.
+ Z.
h
h(r) for all r
then lira inf
Then
be real valued functions on M.
(a)
o
<
M- E.
Let g, h, k and
LEMMA 6.12.
(E)
for j
t.3
g(rj)
g(r.)
3
h(r.).
<_
_<
.
By (6.28),
r.j
Take a sequence
M[t.,l)3
E
Therefore
lim inf h(r.) _< lim sup h(r)
3
Mr+b
j-+oo
q.e.d.
VALUE DISTRIBUTION ON POLYDISCS
663
+
n(b)
Let g
LEMMA 6.13.
Define
IR be an increasing function.
+
sup{g(r)
S
r
In()} -< oo.
e
Then
g(r)
PROOF.
r
.
T <
g(r0).
g(rb) of r
The function
T < g(r
s for
A point r 0 with 0 -< r0 < b exists such that
Hence S -< s.
-< s.
g(r0b)
Therefore g(rb)
increases.
A number r0 e (0, I) exists such that r 0 <
-<
0)
(0,i)
6
Take T < S.
Obviously s -< S.
with r e M.
for r
S
rob.
Then
S and g(rb)
We find that s
S for r
if 0 < r < I.
Again take T < S.
Take r e
r
M[r0,1).
>
tb >
rob
e
IR(0,1)
x) with r0 < t <
which implies T <
g(r)
<
g(r0b)
e B.
and
> T.
g(r0b)
exists such that
Therefore
< S for all
r
M[r0,1).
b if r e M; q.e.d.
S for r
Hence g(r)
r0(T)
0
x + t(h
Then r
t)x + tb
(I
A number r
In order to eliminate Q we need the following well-known Lemma.
(Hayman [123 Lemma 2.4 p. 38)
LEMMA 6.14.
T
[t0,1)
Take t
T(t) -> c for all
R[to,l).
t
lR[t0,1)
t) dt
(i/I
with
Define
> 2T(t)
g
Then E is open in
Let
[0,i).
Assume that there is a
+IR be a continuous, increasing function.
positive number c > 0 such that
6
o
(6.31)
2.
E
Define Q by (6.29).
PROPOSITION 6.15.
increasing function.
T(r)
> c for r e
M +IR
Define T
O
6
/IR be a
continuous,
Max(c,T)
as a function on M.
A continuous
is defined by
(t + (IeTo(t)
p(r)
for all r
M
Assume that there are constants c > 0 and s e IR(0,1) such that
M[s,l).
n
function p
Let T
M where (t,)
R(0,1)
c
)
B is uniquely determined by r
(6.32)
y(t,).
Then
+
r < p(r) < b for r
M and p(r)
n(h).
Let Q(r) be the largest number such that
664
W. STOLL
@(r)r -< p(r) for r
Then @(r) > I.
M.
2@(r)
@(r)
log
for all r
M[s,l)
Define
Y0
g(Sllll2/c)"
=0
log T(r) + 2 log
<
II h rll
Then
+ co
(6.33)
+ cO
(6.34)
In particular
Q.
2@(r)
@(r)
log
on M.
Define
<
x[I
Min{llh
log T(r) + 2 log
x
e
}
> 0 and
B(B)YP(
Y3
2
+ lg
Then there exists an open subset E of M with
T(p())
2T(r)
_<
(E)
ls)"
<
(6.35)
<
3
such that
(6.36)
M- E.
for all
In particular T((r)) -< 2T(r) on M.
PROOF.
r
Trivially T O is continuous and T
O
y(t,) with 0 <
t
<
B.
and g
M.
Take
M.
Then
Hence
+ (I
t)c
<
er(r)
0 < t < t
0 < r
_> c on
y(t,x) < p(r) < y(t,x) < b.
+
Consequently, @(r) >
with x
and p(r)
B and s _< t < i.
p(r)
For each j
E
[l,n]
x +
n(b).
Take r
M[s,l).
Then r
x + t(b
Also
It
+ (I-)
t)c) (b
x)
+ (1
t)c
(b- x).
er(r)
we have
@(r)r.j
with equality for at least one j.
<
rj +
Since
(I-.)c (bj
x.)
r. > 0 we obtain
x)
VALUE DISTRIBUTION ON POLYDISCS
@(r)
We have b- r
t)c
+ (ieT(r)
i
(I- t)(b- x).
ii
where
rj
x. + t(bo
-1
x.)
3
s(b
3
(I- t)x. + tb
Also r.
x.).
Min
b. -x.
J
3
l-<j -<n
3
r.
M[s,I)
lib- x li <-IIII,
@(r)
+
Q.
se
tb
>_
and
3
(6.38)
ro
3
(6.39)
se
b
j
Hence
j
b. -x.
>_
Min
l_<j -<n
Then
-
t)llb- xll
-_<
se
+
(x)
Since
+
t)bj
(r)
Now assume that r
Hence
J
_<
< (i-
J
(i-
Min
er(r) l_<j<n
Xl
b
j
@(r)
3
c
+
O(r)
(6.37)
ro
l_<j_<n
rll
lib
Hence
Min
665
b
eT(r)
> t ->
(x)
j
(x).
(6.40)
(x) by (6.29).
Consequently
t
we obtain
+
lib
xll 2
I+
eT(r)
(6.41)
eT(r)
Now (6.39) and (6.41) imply
2@(r)
@(r)-
811112
T(r)
sc
or
0 < log
2@(r)
@(r)
log T(r) + 2 log
+ log
811112
sc
for all r
M[s,l)
Q, which proves (6.33).
Now (6.28) and Lemma 6.11 prove (6.34).
W. STOLL
666
{r
The open set E is defined by E
T(p(r)) -< 2T(r) for all r
r((t,g)).
rx(t)
M- E.
e
T((r)) > 2T(r)} u M[0,s).
M
For x
B and t
E
IR(0,1)
Then
define
Define
According to Lemma 6.14 we have
i
t
-<2 +log
i_ s
E(X)
Therefore Lemma 6. I0 implies
q.e.d.
Let f be a non-constant meromorphic function on )(h).
LEMMA 6.16.
Tf(rh,sh)
> 0 if 0 < s < r < i.
If f has a point of indeterminacy at O, then
PROOF.
implies that
Tf(rh,sh)
holomorphic at 0.
implies
almost all z
E
If
>_
If(O)
log r/s > 0.
)<h> and s -< t -< r.
)<h>.
)(I) for almost all z
By continuity f[] is constant for all
)--0
Cont radic t ion
LEMMA 6.17.
sufficiently small.
P%IID<h>
0 if % > 0.
n(h)
that
<h>.
Let
Then
P%()z
Because f(zz) is constant for
Hence f
P0
is constant.
q. e d.
Let f be a non-constant meromorphic function on )(h).
/
q
0 for
P%()
f(z)
)(i) we have
Af[](t)
Hence f[z] is constant on D(t) and therefore on
be the development into homogeneous polynomials at 0.
and z
I, identity (5.14)
Then (5.5) and (5.13) show that
X=O
and z e
> 0 and (5.17)
Hence we can assume that f is
f()
Cn
If(0)
0 for some pair 0 < s < r <
Tf(rb,sh)
0 for s -< t -< r.
Af(tb)
for all
Then
and q > 0 with qh < q.
> g and T(=,g) >- c for all
Then there exist numbers c > 0 and s
M[s,l).
Take
(0,i) such
VALUE DISTRIBUTION ON POLYDISCS
PROOF.
A number q0 e U(q,l) exists such that q < q0 b < b.
x + t(
> 0.
Tf(sb,qo)
By Lemma 6.16, c
r
667
x)
Take r
M[s,l).
6
t)x + tb with x e B and s
(i
U(q0,1).
Take s e
Then
s
Hence r >
< I.
_< t
>
q0 b > q
and
q.e.d.
c > 0
T(s,q0h
T(sb,q)
T(r,q)
Now, the Lemma of the logarithmic derivative follows easily, which constitutes
the main result of this paper.
THEOREM 6.18.
function on 3(b).
Take b
+
+n
U
un(b)
and q
Let M be an approach cone.
log+Idzf/ fl
D<r>
n
Let f be a non-constant meromorphic
log+Tf(r,q)
-< 17
lq[l,n].
Take
+
Then
+
(6.42)
19 log
for r e M.
PROOF.
Then c > 0 and s e
Take q > 0 with qb < q.
r > q and T(r,q)
>_ c
for all r
6
(0,I)
n
M +IR
For T(r,q) define p
M[s,l).
exist such that
by (6.32) and
@(r) as the largest number such that
r < O(r)r
_<
p(r) < b
T(r,q).
Then (6.34) and (6.36) hold for T(r)
positive constants.
-< r < b.
Take
log+,fi/f,ln
_<
8 io
for r
M.
Also we have (6.12).
Let
cj
be
Then
g+T f(O(r)r
q) + 9 log
2@ (r)
1
+
C
D<r>
+
9 log
@(r)
log+(2Tf(r,q))
+
9 log
Tf(r,q)
<
;
8
<
;
17 io
g+Tf (r,q)
where the constant is swallowed by
Now,
and Stoll
20 (r)
g+T f (p(r),q)
<- 8 io
+
19 log
log(i/llb
+
C
+
+ 18 log
c
2
+
r[l
(see
eemma 6.12e)
q.e.d.
it is possible to derive the defect relation along the lines of Vitter
6 ].
5
668
W. STOLL
Naturally, Theorem 6.18 extends immediately to any differentiable
operator
n
A
D=
j=l
where the
A. are bounded
we have to take
Ao
continuous functions.
-j
If we want Df/f to be meromorphic,
as bounded holomorphic functions on
unbounded, if we add a
)(b).
We even could take A.
correction term on the right-hand side in
are bounded holomorphic functions on
mDf/f(r,oo)
log
]D<r>
mi)f/f(r,oo)
if the constant c
J
i+
-< 17n
)(h), then Df/f
Df/fl
is meromorphlc.
2
log+Tf (r,l)
(6.42).
If the A.
We have
lFIDf/fIn + log 2
In<r>
+ 19n log+
in the proof includes n log 2 and n log C where C is an upper
2
bound of the A..
ACKNOWLEDGEMENT.
This paper was supported in part by the National Science Foundation
Grant No. 80-03257.
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i.
abls_,
2.
3.
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