Intenat. J. M.
M.
201
Vol. 6 No. 2 (I783) 201-242
UNBOUNDED FUNCTIONS IN THE UNIT DISC
L.R. SONS
Department of Mathematical Sciences
Northern Illinois University
DeKalb, Illinois 60115 U.S.A.
(Received June 24, 1982)
ABSTRACT. A survey is made of results related to the value distribution of functions
which are meromorphlc or analytic in the unit disc and have unbounded growth according
to some specific growth indicator.
KEF WORDS AND PHRASES.
Value dtribution, unbounded funct2n, evanlinna theory.
CODE. 30?35
1980 THEMATICS SUBJECT CLASSIFICATION
i.
INTRODUCTION
We shall consider functions which are meromorphic or analytic in the unit disc
D
zl
{z
< I}
and have unbounded growth according to some specific growth indica-
We give a survey of known results related to value distribution.
tor.
Initially we
shall consider functions whose growth in the disc makes them correspond to meromorphlc
Then we shall proceed to functions with slo-
functions of finite order in the plane.
wer growth.
We assume the notation of the elementary theory of Nevanlinna theory as
it appears in Chapter One of
2.
Hayman’s book [i].
SECTION TWO
First we consider functions which have their growth determined by the Nevanllnna
characteristic function.
Let
f
llm sup
r-l
where
T(r,f)
D.
be a meromorphlc function in
We define the order
of
T(r,f_)
l...oE-lg(Ilr)
is the Nevanllnna characteristic of
Further we define the lower order
u
of
f
f
by
(2.1)
f
at
r.
We note that
0-<u < +.
by
+
rl
-log(l-r)
(2.2)
L.R. SONS
202
0 < E -< u < +.
and observe that
In this case we have the following relationship between the order of a function
and the order of its derivative function.
Let
THEOREM 2.1,
(2.1).
Let
u’
and [3,p.228]) and
If
f
f’
be the order of
’
defined by
<
f
be the
u
D
for which
(2.1) is posi-
defined by
must assume every value on the Riemann sphere and how frequently
An omitted value
it may assume such values,
THEOREM 2.2.
Thus
’
u (TsuJl [2]
and
(Sons [4]).
is a meromorphic function in
tlve, we ask if
defined by
with order
(2.1), and let
respectively defined by (2,2),
f’
lower orders of f and
D
be a meromorphic function in
f
Let
f
is called a Picard
be a meromorphic function in
D
such that
T(r ,f)
r/l
Let
be a meromorphic function in D with
g
(r,s)
Then
for
f(z)
g(z)
0 (),
(r
/
).
has infinitely many zero points in
D
with two possible exceptions
More preclsely,
g.
.N(r i/(.f-g))
lira sup
-log(l-r)
r/l
with two possible exceptions for
g
Exceptions can indeed occur,
(Tsuji [5] and [3,p,297]).
For
fk(z)
We note that
fk
has order
k- i
k > i
we define the functions
fk
in
D by
exp((l.-z)-k).
and
fk
omits the values zero and infinity.
A closer relationship between the growth of the function and the values assumed
is given below.
T!.EORF 2.3. Let f be
defined by (2.1).
{a
k}
Let
a meromorphic function in
be a set of a-values for
D
f
> 0
with finite order
where
a
is in
E u {(R)}.
Then
(i)
E(l-lakl
+i+
<
for each
e > O;
for each
e > 0
k
(ll)
l(l-lakl)
k
with two possible exceptions for
a;
and
UNBOUNDED FUNCTIONS IN THE UNIT DISC
(ili)
when
f T(r, f) lr)l dr = , we have
1
exceptions for
203
Z(l]ak]) u+l
with two possible
(TsuJi [3, p,204 and 293] and [5]).
a
It can be further shown that if
f
D, then the
is a meromorphic function in
convergence of
T(r,f) (l-r) 7I dr,
;i
for some
7 > 0,
implies the finiteness of the three quantities
N(r,a)(l-r) Y-I dr,
where
u {=}
is in
a
(2.3)
{a
k}
and
E(l-lakl) 7+I
n(r,a)(l-r) Y dr,
is the set of
a-values
for
(2.4)
f.
In fact, it
can be shown that the three quantities in (2.4) converge or diverge simultaneously
regardless of the known behavior of (2.3) (see Nevanlinna [6, Ch. X]).
{a
k}
If the
are the non-zero zero points of
> 0
convergence exponent
E(l-lakl)
If
,
E(l-lakl)
If
=,
<
k
{ak}
of the
it is convenient to define the
follows.
O.
then
then
as
f,
E(l-lakl)+le..
is that number such that
k
k
Z(l-lakl)+l+e..
and
<
k
for any
e > 0.
From THEOREM 2.3 we see
0 <] <
if
is the order of
a
p >- i
Let
f.
We assume
[(l-lakl) p
(l-lakl) p+I
and
k
is not an integer, then
+
p
i
is finite.
be a positive integer such that
k
If
u
according as
p
[] + i, and if
E(l-lakl)U+l
k
<
or
is an integer then
E(l-a ) u+l
k
k
.
p
U
or
In any case,
p-l_< u <u.
Now we define the
Z(l-lakl)
<
,
_canonical product P(z)
then
,
k
P(z)
(i
k=l
If’
Z(l-l.l
k
then
formed with
l_lakl2 ).
l,-akz
{ak}
as below.
If
(2.5)
204
L.R. SONS
P(z)
II
k=l
where
P
P
Further, when
D
is analytic in
k)
P(a
and
0
k}
<
i,
{ak}.
where .V
P.
is the convergence exponent of the
{a
f(z)
Let
s
f
0
fl
which has finite order
be the canonical product
Then
k}
(Tsuji [3, p.227]),
be a meromorphlc function with finite order
u
in
D.
can be expressed in the form
f(z)
where
D
P
f, and let
be the order of
THEOREM 2.5,
Then
Let
i, 2, 3,
D).
(z
be the non-zero zero points of
formed with
k
for
We then have two useful theorems.
THEOREM 2.4. Let f be a meromorphic function in
{a
(2.6)
(2.5) we see
is defined by
IPCz)
Let
(A(k,z)) p)
l-lak 12
A(k,z)
We observe that
(1-.A(k,z))exp(A(k,z)+(A(k,z))2+,,.+ P
and
f2
(fl(z))/(f2(z))’
are analytic functions in
(z
D),
D with order
< e (Tsuji
[3, p.227])
Using an extended notion of the characteristic function M. M. Dzhrbashyan [7] has
introduced another factorization for functions meromorphic in
D.
M. A. Girnyk has studied relationships between the growth of products of the form
(2.6) and their value distribution.
In [8] he has determined the asymptotic behavior
of the logarithm of the modulus of products of the form (2.6) assuming the zeros are
-
all positive and have an asymptotic behavior which is essentially
n(r,i/f)
for
0 < C <
,
0 > 0.
c(1-r)
In [9] Girnyk discusses the asymptotic behavior of the Nevan-
linna characteristic and the logarithm of the modulus of products of the form
the case when the zeros possess an angular density.
(2.6) in
He gives an example to show that
even in the case of nonintegral order, the angular density of the zeros does not uni-
quely determine the asymptotics (so his theorems require some additional restrictions).
UNBOUNDED IJNCTIONS IN THE UNIT DISC
205
The parts (i) and (ii) of THEOREM 2,3 and the relationship among the quantities
(2.4) lead us to define Borel exceptional value as follows.
in
Let
be a meromorphic function in
f
lira sup
k/l
D
and define
IoE
N(r ,a)
Hiog(l-)’
for each
.Borel
exceptional
,v for
f
a < =.
0 <
i), and so
/
a<"
f
if
THEOREM 2.6.
is defined similarly.
Let
Then for every
,
a < " If
for f if
valu
u {}
in
D
N(r,a) < T(r,f) + 0(i),
e <
If
we say
be a meromorphic function in
a
by
in
Ua
By the First Fundamental Theorem of Nevanlinna theory we have
(r
a
a
+,
we say
a
is
a
is a Borel exceptional
f
such that
has order
ua
with at most two exceptions,
e
(c.f,
Sons [i0]).
Define
tinct
N(r,a)
in a similar manner to
a-values of
r+l
<
Let
where we consider only the dis-
Let
f.
lira sup
THEOREM 2.7.
N(r,a)
f
Then for every
log N(r,a)
,,log(iH5
a
be a meromorphic function in
a
K u {}
in
D
f
such that
has order
a
with at most two exceptions,
(c.f.
Sons [I0]).
In the same sphere of ideas as THEOREMS 2.6 and 2.7 results may be obtained (c.f.
Sons [i0]) concerning the counting function for simple zeros, and also one may be obtained involving the counting function for distinct simple and double zeros.
The function
defined in
f
D
by
Borel exceptional values and order
f(z)
exp((l-z)
1/2,
-3/2
has zero and
D
Thus a function defined in
as
with a
Borel exceptional value need not have integer order.
Another type of measure of the values a function assumes is the Nevanlinna defici-
ency
which is defined using a ratio of the counting function
characteristic
THEOREM 2.8.
T.
N
to the Nevanlinna
A transitional result in this direction is below,
Let
{a
k}
be a sequence of non-zero complex numbers with moduli
less than one such that
l(z- I%1)=
,
k
but the convergence exponent
of the
{ak}
satisfies
< +,
Let
P
be a canoni-
L.R. SONS
206
{ak},
cal product formed with
(1)
when
p
#
,
there is a positive constant
lim sup
r+l
where
depends only on
c
P.
be the order of
Then
;
0
(ll)
and let
s
and
c
such that
> c
T(r,P)
p (Sons [II]),
Part (i) is also considered in M.A. Girnyk [8].
If
f
D
is a meromorphic function in
and
a
u {}., then the First
is in
Fundamental Theorem of Nevanlinna theory says
N(r,a) + m(r,a)
This theorem relating the behavior of
T(r,f) + 0(I), (r
re(r ,a)
T(r,f)
to
I).
also can give information
about value distribution.
THEOREM 2.9.
Let
f
be a meromorphic function in
lira
D
such that
+.
T(r,f)
r+l
Then for almost all
a,
llm sup
m(r,a)
T(r-fY
<
og
The above theorem was proved by J.E. Littlewood
1/2
[12].
This theorem and some re-
fated results of L. Ahlfors [13] are stated in W.K. Hayman [14] where examples are
given to show their best possible character.
If
f
is a meromorphlc function in
(a)
g(a,f)
lira inf
r-l
(a) (a,f)
8(a)
8(a,f)
1
lim sup
r+l
lira inf
r/l
(a)
and
a
a.
a
we define:
(r,a)
(r
N(r,a)
f)
S(r,.a)
T(r ,f)
a
and
8(a)
It is clear from the definition of
is a Picard value, then
u {}
is in
lira sup N(__)
T(r, f)r+l
1
is called the .Nvanlinna_ deficien..cy of
ultiplicity of
and if
T(r,f)
D
6(a)
I,
is called the index of
g(a)
that
0 < (a) < I,
Some functions may have a number of
deficient values, but for the class of functions under consideration there are some
restrictions on the number and their size.
207
UNBOUNDED FUNCTIONS IN THE UNIT DISC
THEOPo-! 2,10,
Let
lira sup
Then the set of values
over all such values
a
T (rf)
u {}
in
(a)
8(a)} <
> 0
countable, and summing
is
so THEOREM
<
(Hayman [i])
2,
a
D
is a function defined in
I,
(=,f)
()(a)
for which
a, one gets
.{(a) +
f
such that
Log(i-r’
a
If
D
be a meromorphic function in
f
by
f(z)
exp{(1
z)
-2
},
(O,f)
then
2.10 has two as a best possible bound.
It is natural to ask whether functions exist which have an "arbitrary" assignment
of deficiencies at an arbitrary sequence of complex numbers subject only to the conditions of THEOREM
2.10.
THEORE 2.11.
We state two results in this direction,
Let
be a non-negative real number, and let
{k
be a se-
quence of positive numbers less than or equal to one satisfying
k <2
k--1
Let
{a
k}
function
be a sequence of distinct complex numbers.
f
in
D
with order
such that
(ak,f)
Then there is a meromorphic
>
(k)/4,
k
i, 2, 3
(Krutin [15]).
THEOREM 2.12.
Let
{k
be a sequence of positive numbers such that
[ kl.
k=l
Let
{ak}
be a sequence of distinct complex numbers.
D
which is analytic in
b
{aklk-
i, 2, ...}
such that
(ak’f)
Then there is a function
(b,f)
and
k
0
for
b
f
and
E
(Girnyk [16]).
There are some theorems which relate how a number of specific values are assumed
when the function grows rapidly enough.
THEOREM 2.13.
number of roots in
the
are
a
k
q
Let
f
{z
Izl
be a meromorphic function in
< r}
of the equations
distinct elements of
lim inf
u {}
and
D.
f(z)
ak,
q > 3.
(l-r) logM(r f)
-----Jl0g (ir
Let
p(r)
be the total
k
I, 2,
...,
q
where
If
> 7
(2,7)
L.R. SONS
208
where
is some finite positive number, then
y
(l-r)p(r)
>
according as the
a
lira sup
b
r+l
where
y(q.-l)
b
y(q-2)
or
is an analytic function in
D
are all finite or not.
k
If
f
and (2,7) is replaced by
(l-r) logM(r, f)
lira sup
-log(i-r)
r+l
the same conclusion is obtained, and the theorem is best possible (}layman [17]).
Relationships can be formulated between the deficiencies of a function and the
deficiencies of the derivatives of the function at zero.
THEOREM 2.14.
Let
f
(2.2).
is defined by
be a meromorphic function in
f
for which
> 0
where
Then
n+l
If
D
I
Ca.,f)
(0,fCn) ),
D
for which
(a ,f) -<
(0,f(n)),
is an analytic function in
7.
<
j
a.
n > i.
for
> 0, then
for
n > I.
(c.f. Sons) [4]).
Analogous results for functions meromorphic in the plane or for entire functions
appear in H. Wittich [18].
If the functions grow rapidly enough, some conclusions can also be made about the
value distribution of the derivative functions or combinations of the function and its
derivative functions.
The next two theorems give important growth interrelations used
to obtain such results.
THEOREM 2.15.
Let
f
be a meromorphlc function in
T(r,f)
sup
Let
o(-r5--
,(z)
T(r,a i)
i
(i
u)
0, i,
o(T(r,f)), (r
+
I).
in
D
by
k(z) f(kl(z)
is a meromorphic function in
Then
m(r,/f)
and
(2.s)
a
k=0
a
such that
+"
be a positive integer, and define the function
9
where
D
o(T(r,f)),
(r
i),
D
such that
UNBOUNDED FUNCTIONS IN THE UNIT DISC
<
TCr,)
THEOREM 2.16.
Let
f
(+l)TCr,f) + oCT(r,f)),
i__)
T(r f) < N(r,f) + N(r l/f) + N(r,
where in
N (r,i/’)
(z)
are to be considered.
1
N
4-1
o
(r,
+
i),
Then
,) + o(T(r, f)
not corresponding to the repeated roots of
only zeros of
o
is not constant,
#
(r
and assume (2.8) holds.
D,
be a meromorphic function in
be as in THEORE} 2.13, and assume
Let
209
We proceed now to theorems about the values derivative functions assume. THEO_RES
2.15-2.19 are all analogues of theorems in Hayman [19] and/or Hayman [i].
THEOREM 2.17.
Let
f
D
be a meromorphic function in
such that (2.8) holds.
Suppose
o(T(r,f)), (r
N(r,f)
+
I),
and
o(T(r,f)), (r
N(r,1/f)
Let
be defined as in THEOREM 2.15.
I).
is identically constant or
Then
every finite complex value except possibly zero infinitely often.
THEOREI! 2.18.
Let
f
be a meromorphic function in
be a positive integer
f
()
D
such that (2.8) holds.
Then
I )(a, f())
a#
In particular,
In fact, in the
has no finite deficient values except possibly zero.
latter case,
Let
assumes
< i
+
l-!+i
assumes every finite value with at most one exception infinitely
often.
THEOREM 2.19.
If
Let
f
be a meromorphlc function in
D
such that (2.8) holds.
is a positive integer, then
T(r f)
(2 +
--l)N(r
I/f)+(2 +
2)(r
i
f
Consequently, either
f
()
+o (T (r f)
(r/l).
-i
assumes every finite value infinitely often or
f
()
assumes
every finite value except possibly zero infinitely often.
Meromorphic functions with order
e <
+
cients of certain differential equations in
and A.A.
Gol’dberg [24,25]
meromorphic functions in
in
D.
D
have been considered as coeffi-
G. Valiron [20], S, Bank [21,22,23],
have studied restrictions on the growth of the analytic or
D
which are solutions of some differential equations for
which the coefficients are analytic or meromorphic functions of finite order in
D.
L.R. SONS
210
Another gauge of values assumed is the Vallron deficiency.
phic function in
D
(a)
and
a
(a,f)
llm sup
r/l
(a)
is called the
u {(R))
is in
of
f
is a meromor-
we define
T(r,f)
Valiron deficiency
If
T(r,f)
r+l
The next theorem gives some measure of
a.
the size of the set of Valiron deficient values.
THEOREM 2.20.
Let
be a meromorphic function in
f
lira
T(r,f)
+ =.
N(r,a)
z,
D
Assume
rl
Then
i=
r+l
T (r,)
Thus the Vallron
except for at most a set of a-values of vanishing inner capacity.
deficiency
A(a,f)
is positive at most for a set of a-values of inner capacity
zero,
(Nevanlinna [6, p.280] ).
From the definitions of Nevanllnna and Vallron deficiency it is clear that each
Nevanllnna deficient value is also a Vallron deficient value.
How are Borel excep-
tlonal values related to deficient values?
THEOREM 2.21.
Let
f
be a meromorphlc function in
D
which has finite order
and lower order
(1)
If
a
is a Borel exceptional value for
f, then
(li)
If
a
is a Borel exceptional value for
f
(a,f)
and
,
i.
(a,f)
then
i.
(Sons [10]).
Another type of function behavior of special interest is the presence of radial
A function such as
limits, or asymptotic values in general.
f
defined in
D
by
/Z+z
has radial limits 1 and
respectively.
+(R)
as
Izl
/
1
along the negative axis and positive axis
However, we have the following theorem.
THEOREM 2.22,
There exist functions meromorphic in
D
which have no asymptotic
values, finite or infinite (and hence no radial limits). Further,
characterlsitc of such functions may have arbitrarily sl growth.
the Nevanlinna
(MmcLane [26]),
(0. Lehto and K. Virtanen [27] have shown that there even exist meromorphic functions f in D without asymptotic values which are also normal functions in D
211
UNBOUNDED FUNCTIONS IN THE UNIT DISC
that is, they satisfy
for
z
and some constant
D
in
C)0
Meromorphic functions in D which have no asymptotic values possess interesting
value distribution properties,
A function meromorphic in D which has no asymptotic values
THEOREM 2.23.
assumes every value infinitely often.
I-I
o.
Further, in every neighborhood of each point
an
perhaps two values (c.f. Cartwright
[28,29] ).
and Collingwood
The exceptional values in the theorem can occur even in such a way that the two
values cluster only to one boundary point (Barth and Schneider [30]).
K. Barth [31] introduced the class
asymptotic values at a dense set on
D with order
in
f
Am of meromorphic functions
{zllz
i}.
If
f
in
D which have
is a meromorphlc function
u < 2, then the following theorem gives a sufficient condition for
Am
to be in
THEOREM 2.24.
Let
D which is not constant.
be a meromorphlc function in
f
If
1
(1-r)T(r,f)dr
<
0
and for some
a
in
E v {(R)}
N(r,a)
then
f
A
is in
m
0(i),
(t-+l).,
(2.9)
(Barth [31]).
There are examples (Barth [31]) to show condition (2.9) cannot be relaxed to
(a,f)
1.
Let
be a non-constant meromorphic function in
f
{D(e),al associated with the
D(e),
one for each
(i)
D(e)
(ii)
0 < e
e >
finite complex number
O, such that
is a component of the open set
I
< e
2
implies
a
D(el) c D(e2)
{zlzDu{z]
and
D.
An
.a.symptot.i_c trac____t
is a set of non-vold domains
L.R. SONS
212
J
D()
(iii)
n
If
is replaced by
a
the only change is to replace
If(z)-al<e
in (i) by
>
lYne set
>0
{zl
is a non-void, connected, closed subset of
of the tract.
If
K
K
Let
If there exist
a
and
f
b
be a meromorphlc function in
in
u {}
with
N(r,a)
0(i),
N(r,b)
0(i),
b
a
K
Barth [31] has given a suffi
clent condition for the nonexistence of arc-tracts for functions in
THEOREM 2.25.
en_d
is a point, then the tract will be termed a polnt-tract; if
arc, then the tract will be called an arc-tract.
is an
is defined to be the
D
A
such that
m
f
is in
Am
such that
and
then
f
(r+l),
has no arc-tracts.
Another interesting question concerning
A
is whether the derivative of a func-
tion in the class is again in the class.
THEOREM 2.26.
Let
f
be a meromorphlc function in
N(r, )
0(i),
D.
If
(r+l),
and
I
(l-r)T(r,f)dr
<
0
then for each positive integer
and Schneider [32]),
n,
f(n)
is in
Am
or
f(n)
is a constant
(Barth
UNBOUNDED FUNCTIONS IN THE UNIT DISC
213
SECTION THREE.
3.
D with growth measured by the maxi-
We now consider only analytic functions in
mum modulus of the function on circles centered at the origin,
For such functions
there are some types of theorems which are not available for meromorphic functions.
We define
Further, sometimes one can make added assertions for analytic functions,
the M-order of such functions below,
Let
be an analytic function in the unit disk
f
lira sup
O
where
M(r,f)
M-order of
and observe that
Let
(3.1)
-log(l-r)
is the maxlmummodulus of
f
D.
f
0 -< 0 <
{zllz
on
+.
-r},
We say
p
the
is
We define the lower M-order
A
of
f
by
lira inf
r-l
io+lg+M(r’f-)
(3,2)
-log(l-r)
According to M-order it can be shown that a function and its derivative function
We have
behave similarly.
.
THEOREM 3.1.
Let
and lower M-order
order
X’
then
f
f’
If the derivative function
p’
p
be an analytic function in
and
X’
X
For an analytic function in D
(c f
D
such that
has M-order
f
’
has M-order
and lower M-
Boas [33, p 13] and Sons [4])
it is possible to relate the growth of the func
tion to the growth of the coefficients in its power series expansion about zero,
THEOREM 3.2.
Let
f
.
be an analytic function in
fCz)
anZ
D with finite M-order
(zD),
then
-0
p
+ i
limsup
log n
and
0
lim sup
If
0.
(3.3)
log+log+l an
..og.- o’o’1%1_
(MacLane [34, p. 38].
THEOREM 3.3.
If
f(z)
Let
f
be an analytic function in
has the form (3.3),
I+A
then
log n
D with finite lower M-order
214
L.R. SONS
where equality holds if there is an
such that
n
o
Jan/an+l[
is nondecreasing for
nn;
o
(iii)
+
1
log(n-
> llm inf
log+log+[ an
log n
(Kapoor
[35] and Kapoor and
JuneJa [36]).
A theorem of a slightly different nature comes by considering the maximum term
and central index in the power series expansion.
THEOREM 3.4.
0 < O <
Let
f
be an analytic function in
A.
and lower M-order
the maximum term in (3.3) for
Suppose
[z[
{z
f(z)
r}
and
D with M-order
has the form (3.3).
v(r)
p
Let
such that
B(r)
be its central index.
be
Then
lg+l+U (r)
(i)
lim { sup)
r/l inf"
(ll)
llm
(iii)
lim inf
-log(l-r)
r-l
-log(l-r)
log u(r)
1
suP,log (l-r)
The function defined in
+
p
and
(Sons [37]).
D
by
ekjz (k’)3
where
k
o
is
2
and
(kj)
kj+1
2
shows equality need not hold fn the last statement
A well known inequality relates the Nevanllnna characteristic of a function to
the maximum modulus.
T(r,f)
<
If
f
is an analytic function in
R + r
R- r T(R,f),
log+M(r,f)
D, then
(0<r<R<l)
(3 4)
THEOREM 3.5 is a direct consequence of this inequality.
THEOREM 3.5.
a
is the order of
Let
f
f
in
be an analytic function in
D
D
as defined by (2.1), then
with finite M-order
p.
If
a < p < a + i.
Using the notation of THEOREM 3.5 we observe that the functions
fk
defined in
UNBOUNDED FONCTIONS IN THE UNIT DISC
D for
k > i
215
by
fk(z)
have M-order
k
p
exp((l-z)
and order
-k)
To see that other posslbilltles allowed
k- 1.
a
by the theorem can be assumed, we give two theorems.
THEOREM 3.6.
Let
be a convex function defined for
r
0 < r < 1
in
and
satisfying
.(r)
and
_--log(llr
Then there exists a function
defined for
r
Let
and
0
r < i,
in
D
analytic in
S(r,i/f)
T(r,f)
THEOREM 3.7.
f
(.r).
log
lira sup
r+l
< +"
-log(1-r
such that
@(r),
log M(r,f)
(Shea[38]).
(r-l),
be a non-negatlve, increasing functions which are
log r
which are convex with respect to
in
0 < r < I,
and which satisfy
(r)
"log(i--r’)’
r+l
lira
+’
(r)
-ig(l-r
r+l
+’
lim
(r+l),
o((r)),
(r)
r
b(r) (l-r) +
o((r)),
(t)dt
(r+l),
0
and
$’ (r)(l-r)
is monotonically increasing for
there is a function
analytic in
f
log M(r,f)
D
,(r)
in the interval
r
(0,1).
Then
such that
and
T(r,f)
(r),
(r+l),
(Linden [39]).
Turning to a consideration of values assumed by our functions, we see:
THEOREM 3.8.
If
f
THEOREM 2.2 applies and
in
D with M-order
p
D with M-order
is an analytic function in
f
omits at most one value.
I, then f may
(3.4) to get the implication
r/l
is an analytic function
f
omit infinitely many values.
The first part of THEOREM 3.8 follows from setting
llm sup
If
0 > i, then
log T(r
l
liog(l_r
>
O.
R
r
1
+ (l-r)
in inequality
L.R. SONS
216
To see the second part we note that
has M-order
Ig(z)
and
1
p
> 1
g
D
defined in
for
z
D.
in
Translating THEOREM 2.6 fo= an analytic function
can have at most one Borel exceptional value.
exp ((l-z)
-3/2)
by
f
D
in
In fact,
shows that such an
defined by
f
has zero as a Borel exceptional value and M-order O
D
tion also shows that the M-order of a function analytic in
al value need not be an integer.
f
f(z)
3/2.
This func-
with a Borel exception-
D
Indeed it can be shown that analytic functions in
with a Borel exceptional value need not have regular growth according to either order
or M-order.
THEOREM 3.9.
,
lower order
M-order
0,
and lower M-order
f(z)
for
z
D where
in
f
There exist analytic functions
g
D
in
which have order
such that
exp(s(z)),
is an analytic function in
D,
and both
(Sons [lO]).
By using a canonical product which is essentially that of
TWO we obtain a factorlzatlon of functions analytic in
M.
TsuJl
in SECTION
D when the M-order is large
enoush.
THEOREM 3.10.
Let
I <
<
{a
than
[0], the function p defined by
and let
f
be the zeros of
n
p(z)
is analytic in
where
p
I.
Let
f
p
in
D.
O
If
s
is an integer not less
(Linden [40]).
be an analytic function in
D with finite M-order
p
Then
f(z)
where
f
where
an(an-Z )
,= (-)
D and of M-order at most
THEOREM 3.11.
D with M-order
be an analytic function in
p(z)q(z),
is of the form (3.5) and formed using the zeros of
f, and both
p
and
q
UNBOUNDED FIINCTIONS IN THE UNIT DISC
p
are analytic and of M-order at most
(Linden [40]),
D,
in
217
Making additional hypotheses concerning the zeros of
f, we obtain
a theorem
concerning functions of smaller }. order growth,
THEOREM 3.12,
Let
O.
{a
Let
be the zeros of
n
D which has finite H-order
be an analytic function in
f
f
D, and define n(r,8,f)
in
to be the number of
these zeros satisfying
lan
r<
(l+r)
18-
and
arg a
n
-< w(lr)
Set
n(r,f)
n(r,8,f)
max
and
log+n (r
f)
’log (ir
lira sup
y
0<8<2w
r+l
Then
p(z)
where
is of the form
functions in
Using
If(O)
and
an
r
has finite M-order
f
[y]
and both
p
and
q
are analytic
(Linden [40]).
max(p,y)
it can be seen that if
f
is an analytic function in
M(R f)
log
<
0 < r < R < i.
logIR/r
p, then taking
R
i
--1(l+r)
e > 0
we find for
there is
such that
o
"l--e
n(r,f) < (l-r)
for
r
D
i, then
n(r,f)
If
(3.5) with s
D with M-order at most
Jensen’s formula
(zED),
p(z)q(z),
f(z)
o
< r < i.
If a function has nearly all the zeros this allows, the zeros will
THEOREM 3.13.
(R).
Let
f
such that the number of zeros of
(l-r)
for
r
and any
s
f
8
(0<8<2), there
< r < i
o
p
in
is a number
D where
r
o
in
Izl -<7i
-<
is at most
i} as THEOREM 3.13 shows.
be an analytic function which has M-order
Then for each positive
"p’e
IzI
{z
need to be fairly uniformly distributed near
I -< p <
(3,6)
zl
(Linden [40]),
For smaller M-order we have:
THEOREH 3,14,
Suppose
p < n,
Let
f
be an analytic function in
Then there is a constant
any zegion
{z[[sin
i
(Earg z)
D which has M-. order
A such that the number of zeros of
<
(l-r)"
p < i.
f
in
L.R. SONS
218
A/(l-r)
is less than
0 <- r < I
whenever
and
0 < 8 < 2
(Linden [41]),
If restrictions are made regarding the location of the zeros of a function, the
bound in (3.6) can be made better.
Let
THEOREM 3,15.
e > 0
and
q
D
be an analytic function in
f
Suppose that the zeros of
Then if
THEORI24 3.15 gives an example.
f
which has finite M-order
all lie on a finite number of radii of
max(p+e, I),
there is a constant
C
.z zl
i}.
such that
-q
n(r,f) < C(l-r)
0 < r < I
whenever
(Linden [41]).
In general for functions with order less than one, it is not possible to improve
(3.6) much.
In particular, if we let
and let
B
D
be defined in
i
i
ak=
(k=l,2,...),
2
k(log k+2)
by
B(z)-k=l
then
B
is an analytic function in
n(r,1/B)
z
D
which has M-order zero and
1
(l-r) { log(l-r) }
(r/l)
2
(c.f. Linden [41]).
Turning our attention to values assumed by the derivative function, we shall give
two theorems.
THEOREM 3.16.
If
f
is an anlaytic function in
lira sup
(l-r)log M(r,f)
D with
+=
r+l
then either
f
f
assumes every complex number infinitely often, or every derivative of
assumes every complex number except possibly zero infinitely often (c.f. Hayman [17]
and [19]).
The exceptional position of zero in the above theorem cannot be eliminated by
assuming a stronger growth condition.
Functions
f
defined in
D by
UNBOUNDED FUNCTIONS IN THE UNIT DISC
exp(| exp g(z)dz)
f(z)
where
D
is an arbitrary function which is analytic in
g
f’ (z) # 0
z
for any
in
THEOREM 3.17,
and
So for the first derivative one cannot strengthen
D.
If
f
D
is an analytic function in
with
+
sup (1-r)log log M(r,f)
r+l
assumes every complex number infinitely often (Hayman [19]),
then either
f
or f"
Suppose
f
is an analytic function defined in
E
If
is a positive number.
T
f(z) # 0
satisfy
However, one can prove:
THEOREM 3.16.
where
219
D
by
is a fixed bounded set and
a fixed posi-
rive integer, then
M(r,f)
lira (l-r)log
and if
is sufficiently large,
7
E.
value in
y
fV(z),,.,,f()(z)
f(z),
Thus the growth hypothesis in
for
z
in
V
assume no
THEOREM 3,16 cannot be sharpened (}layman
[19]).
To see the growth hypothesis in THEOREM 3.17 cannot be sharpened, we define
and
g
in
f
D by
f (z)
z
exp([
g()d)
0
where
g(z)
and
is a positive constant.
a
fz) # 0
also
and
f"(z) # 0
f’ (z)
for
0
z
in
Clearly
z
for
D
expCa(l_
in
D.
f
+ -----!----1
I/2 ))
(1.z)
is an analytic function in
For all sufficiently large
a, one can show
(Hayman [19]).
The situation concerning asymptotic values for functlons analytic in
ent from that for meromorphlc functions.
tence of such values.
D with
D
is differ-
We give three theorems concerning the exls-
L.R. SONS
220
THEOREM 3.18.
Let
f
be an analytic function in
D.
Then
f
has at least one
f
need not have a
asymptotic value (MacLane [34, p.7] and Nevanlinna [6, p.292]).
THEOREM 3.19.
Let
f
be an analytic function in
D.
Then
radial limit along any ray (Bagemihl, ErdSs, and Seldel [42, Th.7] and MmcLane [43]).
THEOREM 3.20.
Let
f
D.
be an unbounded analytic function in
Then
f
need not
as an asymptotic value (MacLane [34, p.71]).
have
an analytic function may have more than one asym-
D
At a point on the boundary of
totic value, but it is possible to determine the size of the set of such points.
THEOREM 3.21.
zl
{z
on
i}
Let
f
at which
be an analytic function in
D,
E
Then the set of points
has more than one asymptotic value is at most countable
f
(Bagemihl [44] and Helns [45]).
G. R. MacLane introduced the class
A of non-constant
D
analytic functions in
to be those functions which have asymptotic values at a dense set on
{z
zl
i}.
THEOREM 3.22 gives the limitations on the growth of such functions.
THEOREM 3.22.
Let
f
be a non-constant analytic function in
D.
If
0
then
f e
A.
In partlcular, if
has finite M-order, then
f
c > 0. there exists a non-constant function
A
f
and for some number
r
with
o
0 < r
f
o
which is analytic in
Further, for
D
such that
< i,
e
M(r,f) < exp exp
A.
f
(’l’-r) (-iog(l.-)Y
(r <r<l)
o
(Hornblower [46]).
(G. R. MacLane [47] and P. J. Rippon [48] both give different discussions of
THEOREM 3.22)
MacLane [47] also introduces the classes
D.
functions in
B
Izlffil}
{z
zlffil} and
and such that on each
To define
t
[
of non-constant analytic
is the class such that there is a set of Jordan arcs
each ending at a point of
{z
B and
r
r
in
D,
such that the end points are dense on
either
f
tends to infinity or
f
is bounded.
we first need the notion of a set ending at points of the unit clr-
cle.
Let
S
be a subset of
D.
For each
r
with
0 < r < 1,
label the componente of
UNBOUNDED FUNCEIONS IN THE UNIT DISC
(zJr<Jzl<l)
S
dimaeter of
S
Skit)
by
k(r),
here
k
d(r)
(zlJzl
i)
d(r)
if and only if
(z[ lf(z)
every level set
k(r)
Sk(r).
We say
is that
A=B=Lp
approaches one.
f
(In [31] K.F. Barth defined the corresponding classes
constant eromorphlc functions in
A
each other and from
m
F
D.
can be mapped onto
the map and
F
Izl
m
z
o
in
and
m
of non-
of
D
may have infinitely many asymptotic
Such a function was noted in [34] by considering
i}.
Grosz [49] for which every complex number is an asymp-
W.
The function
A
is a function in
value at one point
lzl
B and
F
If the plane is sllt along one path on which
totic value.
for ich
as defined in SECTION TO).
{z
the entire function
D
D, but the classes turn out to be distinct from
Actually, a function which is analytic in
values at a point on
(z
in
so a non-constant analytlc function
THEOREH 3.22 is also in
D with finite H-order by
in
0, ends at points of
ends at points of
S
r
decreases to zero as
},
LcLane’s startllng result
f
sup d
k
is then the class of non, constant analytic functions
l
be the
and set
if there are no components
0
dkr)
Let
is a non-negative integer.
d(r)
with
221
z
with
f
/
=,
then the sllt plane
D by means of the composition of
defined in
which has every complex number as an asymptotic
i.
o
THEOREM 3.23.tells us that the number of asymptotic tracts at a point for a function analytic in
THEOREM 3.23.
is a number
y
D
is related to the order of growth of the function.
Let
with
f
be a non-constant analytic function in
y > 0
D.
Suppose there
such that
i
(l-r)71og
M(r,f)dr <
0
Let
z
o
be a complex number such that
of distinct asymptotic tracts of
values are finite (infinite).
f
z
o
i, and let
associated with
z
o
nz’n(R))
for which the
Then
nf
<
[7 + I] and
n
< [7
denote the number
+ 23
asymptotic
L.R. SONS
222
[34,
(Drasin [50] and IfcLane
is the usual greatest integer function.
where
p.54]).
THEOREM 3.23 is best possible.
D
fined in
To see this is true for
I
n_
0 < y < i,
and
2
n
f
we consider
at
0 <
z
f
de-
I.
To see the result is best possible when
O
D
l-z
When the number of tracts for
2
l+z
z
by
1+6
(z)
n.
for which
< y < i
+
1
defined in
f (z)
where
consider
by
f(z)
for which
0,
y
and
.l+z.
sinh
I_-)
n
3
1+6
at z
(MacLane [34, p.77-
i
O
D,
is restricted for an analytic function in
it is possible to say more about the set where the function has finite asymptotic
values.
THEOREM 3.24.
Let
Izl
of
be an analytic function in
and the ends of the arc tracts of
many tracts for
{z
f
i}, then the Lebesgue measure of the set
with
II
a
If
f. for
f
has only finitely
do not cover
is positive where
A
is the set
such that
f
has the
for which there is a complex number
i
asymptotic value
A
D.
a
(McMillan [51]).
at
Another growth behavior of interest is how the minimum modulus of a function on a
circle about the origin acts.
THEOREM 3.25.
where
p > i.
Let
f
be an analytic function in
Then there exists a constant
K
D
with finite M-order
such that
log L(r,f) > -K(log M(r,f))(iog log M(r,f))
for a sequence of numbers
r
increasing to one where
L(r,f)
rain
If(z) l,
(3.7)
(Linden [52,53]),
The above theorem is best possible (c.f. Linden
growth there is THEOREM 3.26.
[52,53]).
For functions of lesser
UNBOUNDED FUNCTIONS IN THE UNIT DISC
Let
THEOREM 3.26.
p
where
p < I.
0 < s
with
o
be a non-constant analytic function in
f
such that if
contains a set of values
s
L(r,f)
s
satisfies
< s <
o
K
is defined by
D which has order
the interval
l,
s
and a number
o
(s,l%-(l+s))
for which
lOg(l.lr)
i
(3.7).
C
and
C(ls)
of measure at least
r
log L(r,f) >
where
K
Then there exist positive constants
< i
223
C
The constant
need not be less than
1/4 (Linden
[54]).
D
Finally, we shall look at the subset of
in which a given function has its
maximum modulus greater than one and observe how this set intersects
for
0 < r < i.
Let
{z
0 < r < i
For
and let
r8
let
k(r)
(r)
If(z)
For
Let
I
0 < r < i,
_- -
0, then
r+l
If there exist a number
r
o
o(r)
r
o
< r < i, then
THEOREM 3.28.
order
{zllzl
r}
,
lles wholly in
The following two theorems
[55].
D
which has finite
M-order
> 2
0
< r
< i
o
and
2(l-r)
i’+’2p"
I
p
Let
if
Arima in
such that
0 < OCt) <
for
which are contained in
8k(r).
max
k
K.
r}
be an analytic function in
f
lira sup
(ii)
{zl zl
8(r)
and lower M-order
If
D}.
z
and
line of reasoning used by
a
r}
and let
be the arcs of
+=; otherwise set
THEOREM 3.27.
(i)
D,
> i
be their lengths.
D, define 8(r)
follow from
Izl
First we introduce some notation.
be an analytic function in
f
{z
f
defined by (i.I).
D which has finite positive
be an analytic function in
If there exists a number
0 < O(r) -<
2T
2+2y
(l-r)
r
o
such that
224
L.R. SONS
for
r
then
< r < i,
o
Some subclasses of finite M-order have received special study.
is an analytic function in
f
If
IfCz)l
C
where
D,
f
-< =p
D,
is said to be in
(z
.
If
0 < n <
(0 < p < )
I
zl
B -n
is said to be in
(cC-Izl>-n>,
is a positive constant and
Bp
f
f
if and only if
is an analytic function in
if and only if
(.og+lf(z)[) p
dx dy < =,.
<i
E. Belier [56] points out that these classes are related by the following inclusions:
(l/n) +
B-n
for all
> 0
0 < n < ;
and
B
p
c
5-2/P
(I
p < ).
Beller [56,57] has proved the following theorem concerning the zeros of functions in
these classes.
THEOREM 3.29.
0 < n <
and
{a
(i)
k}
If
f
is an analytic function in
are the zeros of
[
k:l
(ii)
If
f
k}
are the zeros of
in
D, then for
(-[akl) "++
is an analytic function in
and the {a
f
f
D
in
which is in
e >
-n
O,
<
B
which is in
D,
D
p
where
i < p <
e > 0,
then for
1+1/p+
I
(1-1:1)
k=l
For
B
I
we get further information by taking
0
in the theorem of
Heilper [58] which follows.
THEOREM 3.30.
If
f
is an analytic function in
D
with
] (log+[f(z),)(1-1z,)
dx dy<=,
A.
where
225
UNBOUNDED FUNCTIONS IN THE UNIT DISC
> 0
for some real number
{a
k}
and
are the zeros of
f,
then
k=l
If
0 < u < i, the condition is sufficient,
[59] showed that for some n,
H,S. Shapiro and A,L, Shields
in
B
-n
{ak} are the zeros of
and
I (l-lakl)
<
f
if
f
is a function
on a single radius of the unit circle, then
theorem was extended by W.K, Hayman and B. Korenblum [60]:
Their
k=l
THEOREq!
3.31.
[0,i) such that
is a positive, continuous non. decreaslng function of
If
(r)
approaches infinity as
D
analytic function in
{a
k}
approaches one, and if
f
on
is an
for which
loglf(z)
then those zeros
r
r
of
f
.
<
(Izl),
(zD)
which lie on a single radius of the unit circle satisfy
(i-
lakl)
(3.8)
<
if and only if
0
(3.9)
dr<(R)
C.N. Linden [61] has found additional conditions on
in the previous theorem
besides (3.9) to conclude that (3.8) holds for the zeros of
D.
regions in
D
and the behavior of its zeros which lie on a single radius of
the unit circle and also has some interesting
(1)
If
xamples
concerning their best-posslble
He shows
THEOREM 3.32.
zeros of
in certain tangential
Bruce Hanson [62] has shown further relations between the growth of an
analytic function in
nature.
f
f
Let
f
be an analytic function in
which lle on a single radius of the unit
0 < A <
D,
clrcl.e.
and if
I
(log+M(r,f))(l_r)
0
then
I (l-lak I)+I
k--1
< ="
dr <
,
and let
{a
k}
be the
L.R. SONS
226
(il) If
01 log+M(r,
f) dr <
then
Another interesting class of functions in D
is those defined by gap power
If the gaps are large and regular, the conclusions on value distribution are
series.
especially nice even for functions which grow slowly:
THEOREM 3.33.
Let
f
D
be a non-constant analytic function in
defined by the
power series representation
Z
f (z)
k=0
{n
k}
where the sequence
satisfies
nk+l
for some real number
(i)
(3.10)
CkZ
>-q
> i,
(3.11)
(k=1,2,3,...)
q.
If
then
f
assumes every complex number infinitely often.
every complex number infinitely often in every sector in
{-.Io,
a and 8 are in
where
<
0 <
z <
Is[
In fact,
D
f
assumes
of the form
<
[-2, 2] (Mra [63]),
(li)
llm sup
k-
Ckl
> 0
then
(a,f)
for all
a
(?(urai [64]).
in
THEOREM 3.34.
Let
f
power series representation
(i)
f
(ii)
if
is in
be a non-constant analytic function in
(3.10
where the sequence
A;
llm sup
k
then
f
0
..[Ck[
> 0
has no finite asymptotic values (Murai [65]).
{nk}
D
defined by the
satisfies (3.11).
Then
UNBOUNDED FUNCTIONS IN THE UNIT DISC
227
For smaller gaps in the series representation for the function, there are also
value distribution results
ones relating the gap size and the growth of the func-
tion.
THEOREM 3.35.
order
{}
Let
f
D
be a non-constant analytic function in
which has M-
and is defined by the power series representation (3.10) where the sequence
0
satisfies
lira inf
k
lg(nk+l -_nk)_
log nk+I
>
I ((3.12)
)
Then
(i)
for each complex number a,
(a,f)
0;
lira sup (L(r,f)/M(r,f))
(ii)
where
L(r,f)
i,
(3.7)
is defined by
and
(iii) f
has no finite asymptotic values (Wiman [66]).
In [67] the conclusion (i) of THEOREM 3.35 is obtained for some functions with
less restrictive gaps than (3.12).
A result where the density of the sequence
{nk}
and the growth of the function is linked follows.
THEOREM 3.36.
Let
f
be a non-constant analytic function in
power series representation (3.10).
Let
and suppose that, for some fixed
t,
(t)
If the M-order
p
of
f
(t)
be the number of
D
n
k
defined by the
not greater than
8(0<8<1),
0(tl-8),
(t-).
satisfies
(3.13)
8
then
(1)
(a,f)
0
for every complex number
a, (Nicholls and Sons [68])-
and
(ii)
f
has no finite asymptotic values (Sons [69]).
The best possible nature of (3,13) to obtain (li) is s[own for
p
k
where
p
is an integer not less than 2
by examples in [70]
n
k
of the form
L.R. SONS
228
4.
SECTION FOUR.
In this section we give some theorems concerning functions meromorphic in
with growth more restrictive than that in SECTION TWO,
Define
and
a
D
by
(r,f)
lira sup
and
lira inf
r+l
where
T(r,f)
.T_(r,f)__
(4.2)
-log(l-r)
is the Nevalinm characteristic of
f
r.
at
Then
0 N
N
N
+.
It is possible to obtain some restriction on the growth of the derivative of a
function in this class in terms of the growth of the function.
Let
IF.OREN 4.1.
and
<
+.
be a meromorphic function in
f
Then
lira sup
r-l
If
f
defined by (4.1)
D with
T (r,_f’)
_o,_r
< 2(+i).
D, then
is in fact analytic in
li sup
r/l
T (r,f’)
(Sons [4]).
=+2.
.-.rrl-
The growth of the function places some restriction on the number of a-values and
f
placement in the disc where
Let
THEOREM 4.2.
and
<
4.
{ak}
Let
may assume a-values.
be a meromorphic function in
f
be the set of a-values for
f
D
defined by (4.1)
with
where
a
is in
Then
(1)
for each
(ii)
the product
>
0,
I (1-1%1)
k
P(z)
1+
<
m;
.-]a.I
H (1- "i; E z
2
) exp
n
converges for each
(iii)
for each
e > 0
z
in
r+l
rC
2
n
D;
there exists a constant
lira sup
1-,la.I
-_
E z
K
such that
(l-r)l+l+(i/IP(r)J),
-log(l-r)
K.
E u {}.
UNBOUNDED FUNCTIONS IN THE UNIT DISC
229
4
for
i, 2,..,. (TsuJi [3, p.204]).
n
A restriction on the possible number of zeros (or a-values) is Riven bel.
TflEOR/{
4.3.
Let
be a meromorDhlc function in
f
defined by (4,1),
D
for which
0 < a < 4o with
Then
(l-r)n(r)
(Sons [4]>.
r/l
We now consider the number of values a function in our class can omit and number
of times values are assued relative to the growth of the function.
Let
f
be a meromorphic function in
llm sup
D,
and define
for each
a
-log (l-r)
a
N(r,a) < T(r,f) + 0(i),
By the First Fundamental Theorem of Nevanlinna theory we have
0 <
THEOREM 4.4.
Ua
Let
-<
B(R)
f
defined by (4.1).
with
tions, where
by
in
N (r ,a)
r/l
(r/l), and so
a
is defined similarly.
be a meromorphlc function in
Then for every
a
>
for which
u {(R)}
in
q > 3,
a
D
(l/q) ((q-2)
-
with at most
q-i
excep-
(c.f. [i0]).
i).
In SECTION TWO we introduced the Nevanlinna deficiency and the index of multipliUsing the notation established there, we have the following theorem related to
city.
THEOREM 2. i0.
TEOREM 4.5.
f
be a meromorphic function in
is defined by (4.2).
where
)(a)
Let
> 0
D
Then the set of values a
such that
u {(R)}
in
for which
is countable,
[
{(a)+0(a)} <
[O(a)
< 2
+!
(Sons [4])
a
An immediate corollary to THEORKM 4.5 presents a limitation on the number of
values such a function can take only a finite number of times.
THEOREMS. 4.6.
where
Let
f
be a meromorphic function in
is defined by (4,2),
If
>
/(p-2),
D
such that
L.R. SONS
230
where
is an integer larger than
p
2, then f
cannot have
p
values taken only a
finite number of tlmes.
In fact, it is true that the above theorem holds if
is replaced by
.
We
have
IEORK._’
where
4.6’.
Let
f
be a mero:rphic function in
(4.1).
is defined by
D
such that
If
u > i/(p-2),
where
cannot have
p
values taken only a
rely on the fact that when such
f
take on a value
is an integer larger than 2, then
p
finite number of times (Tsujl
THEOREMS 4.6 and 4.6’
[3, p.215]),
(a)=l.
only a finite number of times, then
each integer
p
with
f
p -> 3, we Consider an automorphic function
I, a 2, a3,...,aP
and [71, p.153]),
a
For
For
These theorems are best possible.
which maps the
g
punctured at the
unit disc onto the universal converlng surface of the plane
points
a
p
i/(p-2), (Nevanlinna [6, p, 272]
g we have
An immediate corollary to THEOREM 4.6’ is the following.
THEOREM 4.7.
where
e
Let
is defined by
f
be a meromorphic function in
(4.1).
Then
D
such that
0 <
<
+
cannot omit an infinite number of values.
f
The techniques used in proving THEORE_S 2.17 and 2.18 make it possible to estab-
llsh some theorems concerning the value distribution of derivatives of functions in
our class.
THEORF21 4.8.
where
Let
is defined by
f
be a meromorphlc" function in
(4.1).
D
such that
3 < s <
+
If
N(r,f)
o
(T(r,f)),
(rl)
and
NCr,i/f)
then
fv
o
(TCr,f)),
(r+l),
assumes every finite complex value except possibly zero infinitely often
(Sot= [4]).
If one is willlng to take
order derivatives.
e
larger, slmilar concluslons can be made for hlgher
231
UNBOUNDED FUNCTIONS IN THE UNIT DISC
An analogue of THEOPM 2.18 is obtained by consideration
of the lower growth
of the derivatives.
EORE_
where
Let
4.9.
f
(4.1),
is defined by
function
be a meromorphlc
Let
v
D
in
0 <
such that
<
+
be a positive integer, and let
())_
lim inf r(__r,f
-log (lr)
rl
[(a,f (9))
Then
In particular,
f()
often provided
>
< i
+
i_+ l_!
assumes every finite value with at most one exception infinitely
(+I)/
(Sons [4]).
Turning to consideration
ofasymptotlc values, we observe that THEOREMS 2.24 and
2.26 have the following as a corollary.
TEOREM 4.10.
<
+
Let
f
be a non-constant meromorphlc function in
is defined by (4.1).
where
Suppose there exists an
a
D, with
u {}
in
such
that
N (r,a)
0 (1),
(r/1).
Then
A
(1)
f
(ll)
for
(Barth [31]);
n > 0, either
f
(n)
is in
Am
or
f(n)
is a constant function
(Barth
and Schneider [32]).
If one considers the class
e <
+
where
are normal in
e
is defined by
F of all meromorphlc
(4.1), an important subclass
D,
K
in
D
such that
is those functions which
f
is a normal, meromorphic
then
T(r,f)
where
f
D.
O. Lehto and K. Virtanen [27] have shn that if
function in
functions
< K
is a positive constant,
log(1/(1-r)) + 0(i),
(r-l),
Further, it is known (c.f. Hayman and Storvick [72])
that the derivative of a normal function need not be a normal function, but it cannot
grow too rapidly either according to THEOREM 4.1.
L.R. SONS
232
One interesting property of normal functions is how asymptotic values at points
of the unit circle are related to angular limits at such points.
THEORE 4.11.
If
asymptotic value
a
angular limit
at
a
f
D
is a normal meromorphic function in
along an arc in
D
ending at
,
II-i,
and has the
then
f
has the
(Lehto and Virtanen [27]).
For normal analytic functions another value distribution result can be stated.
THEOREM 4.12.
(-i)
f
Let
f
D,
be a normal analytic function in
has finite radial limits on a dense subset of
{z
Izl
Then either
=i};
or
(ii) f
assumes every complex number infinitely often (Sons [73]),
Other interesting examples of functions in
F
can be found in [42] and
M. Tsuji [74] has shown that certain automorphic functions are in
F.
[59],
Also the deri-
vatlves of some functions of bounded Nevanlinna characteristic are not of bounded characteristic but are in
5.
F
(with e < I) (c.f. Hayman [75]).
SECTION FIVE.
We consider theorems concerning functions analytic in
tive than that in SECTION
THREE.
For an analytic function
D
with
f
in
rowth more
D
define
0
restric-
and
by
lira sup
log+M(r ,f)
-log(iLr)
(5.1)
0,
and
lira inf
where
M(r,f)
is the maximum modulus of
lg+M(r’f)
(5 2)
21og(--l-r)f
on
{zllzl=r}.
We have
0 < ), < p _<..I-.
Subsets of the class here include univalent functions (the classical Koebe function has
0
2), p-valent functions of
various types, and functions in
H
p
classes
(0 < p < ) (c,f. Duren [76], Hayman [77], and Pommerenke [78]),
Some restriction is obtained on the growth of the derivative of the function in
terms of the growth of the function,
THEOREM 5.1.
Let
f
be an analytic function in
D
for which
0
is defined by
UNBOUNDED FUNCTIONS IN THE UNIT DISC
(5.1)
and
f) >
0,
Then
p’
p <
The funtion
233
f
(l-z)
"I
+ i.
<
jlogtl--x
f(z)
D by
defined in
logtM (r ,f’)
lira sup
r/l
has
O’
and
i
O
2.
The growth of functions in this class is reflected by the power series structure
of the functions as follows:
THEOREM 5.2.
(5.1) and
p > 0.
Let
f
D
be an analytic function in
for which
is defined by
O
If
a z
n
f (z)
(zeD)
n--0
then
8 < O,
80
where
lim sup
80
log+l a_n
and 8
lim sup
log n
r/l
-log(1-r)
(c.f. Sons [73]).
The function
f
defined in
D
f(z)
by
(l-z)
has
SO
0
and
p
i,
so
the inequality may occur.
Inequality (3.4) gives (the easy) THEOREM 5.3.
THEOREH 5.3.
(5. i).
where
Let
f
be an analytic function in
D
for which
is defined by
Then
a
is defined by (4.1).
J. Clunie [79] enables one to obtain many
An adaptation of a construction of
functions in the class.
THEOREM 5.4.
Let
be an increasing, convex, continuous function defined on
the interval [0,i) for which
(r) =+.
li=
Then there exists an analytic function
(1)
og (r,f)
(ii)
T{r,f)
(r),
(r),
(r/)
(r+l).
f
in
D with
L.R. SONS
234
JensenVs formula,
Using
satisfying (5,1) with
p <
THEOREM 5.5,
f
Let
defined by (5,1) and p <
one can get the following bound on zeros for a function
,
+
be a function which is analytic in
+
.
D.
Assume
0
is
Then
(l-r)n(r,I/f)
llm sup
rl
(5 3)
< p
-log(l-r)"
An example in [59] shows that the left-hand side of (5.3) may be positive.
THEOREMS 3.15 and3.15 show the zeros of the example must be spread around the disc.
If
f
grows in a regular enough way, some additional results can be obtained.
If
f
is an analytic function in
IfCz)
where
C
<
D,
f
is said to be in
A-n
if and only if
c (l-lzl) -n (zD)
H. Shapiro and A. Shields [59] note
0 < n <
is a positive constant and
that these are precisely the functions for which their Taylor series representation
have
I
n=l
2) /nk
<
k.
for some
If
(I an
f
D, then
is an analytic function in
f
is said to be in
Ap (0 < p < )
if and only if
jj If(z)IPdx
dy <
(For p=2, one gets the "Bergman space").
E. Belier [56]points out that for all positive
sions hold
A-2/p
Ap
and for all
e >
0,
A-n
C
A
(i/n)+e
p
and
n, the following inclu-
UNBOUNDED FUNCTIONS IN THE UNIT DISC
p
For functions in
A
THEORKM 5.6,
f
and if
{a
k}
If
235
the followlng can be proved,
is an analytic function in
are the zeros of
f
D which is also in A
p
(0
<p <
(R))
arranged so that
lal <-la21 la31-<...,
<
then
lira sup
rl
(l-r)n(r,i/f) -< i
log(lr P
and
llm sup
n-
l-lanl
n-llog
-< 1
P
n
Further, those results are sharp (Belier [57] and Horwitz 0]).
In [’81] B. Korenblum has given a necessary condition for a subset of
A-n;
a zero set of
be a zero set of
he has also given a sufficient condition for a subset of
-n
A
u
n>O
f in
A
to be
D
and a necessary and sufficient condition for a subset of
be a zero set of
(Every
D
to
D to
.
has the form
Z
f(z)
k---O
Ck zk
with
c
for some
k
0(kn),
(k),
n).
There are functions defined in
D
by gap power series wlth gaps smaller than
(3.11) and gr#th not as rapid as that implied by (3.13) for xhlch conclusions of the
nature of THEOREM 3.36 can be made.
THEOREM
5.7. Let
f
For example,
be an analytic function in
representation (3.10) where the sequence
{n
k}
D
satisfies
defined by the power series
236
Lo SONS
2
v
n-ns
k--O
If
0 > 0
(i)
where
0
)2,
O((log n
(5,1), then,
is defined by
O;
a, 6(a,f)
for each complex number
in fact
llm sup(_H(r,l /(f-a))/(log H(r,f))
rl
1;
(Nicholls and Sons [68]).
and
(ll) f has no finite asymptotic values (Sons [82]).
6.
SECTION SIX.
In the earlier sections we have considered functions
in
D
T(r,f)
for which either
or
M(r,f)
is unbounded as
of the results stated in those sections apply to functions
than having
u
or
or
u
r
approaches one.
Some
which grow less rapidly
f
(2.1), (3,1),
positive, where these are defined by
or
(4.1), and (5.1) respectively.
analytic or meromorphic
f
There are some other results known for functions of
slow growth.
In [71] R. Nevanlinna indicates that if a function
f
omits an infinite number
of values subject to certain special conditions, then
T(r,f)
0(log log
i- )’
(r/l)o,
The set of Bloch functions, a subset of the set of normal functions introduced in
SECTION TWO, has slow growth.
An analytic function
sup(l-lzl 2 )lf’(z)!
f
in
D
is called Bloch if
<
z6D
We have
TIIEOREM 6.1.
Let
f
be an analytic function in
D which is a Bloch function.
Then
llm sup
rl
N(r,i/f)
log log(l/(l-r))
< lim sup
r/l
T(rtf)
ig log(1/(1-r))
_<
Conversely, there exists a Bloch function such that equality holds in (6.1)
i
y
(6.1)
237
UNBOUNDED FUNCTIONS IN THE UNIT DISC
(Anderson, Clunie, and Pommerenke [83] and Offord [84]),
In recent years annular functions have been studied concerning their value disBonar [86]
tribution (c.f. Bagemlhl and Erds [85]
f
analytic function
D
of Jordan curves in
(i)
J
(ii)
for each
THEOREM 6.2.
as above.
{r
Let
contained in
e > 0
n
{zlIzl
n
Jn+l
there exists a number
Let
f
+
n
such that, for
O
{zIl-e<Izl <I},
lies in the region
min{If(z) llzEJn
n
An
such that
no, Jn
n >
{J
is said to be annular if there exists a sequence
lies in the interior of
n
(ill) (J
D
in
and Bonar and Carroll [87] ).
as
and
-.
n
be an annular function in
D
with defining curves
J
be a sequence of numbers increasing to one such that
< r
n
},
n
{J
n
is
Then
N(r
+/- if
n
tog
l/f)
>
(S
i
(Nicholls and Sons [68]).
Another growth indicator for functions
m (r,f)
2
analytic or meromorphic in
D
is
defined by
2(r,f)
m
{’2
A number of properties of m2(r,f)
m2(r,f)
f
appear in [88].
is
dO) I/2
(0 <r < i).
and some value distribution theorems involving
In [89] some additional results are obtained with special
consideration given to the case where
one, but T(r,f)
(loglf(re I0) I) 2
0
bounded.
m2(r,f)
approaches infinity as
r
approaches
L.R. SONS
238
RE FE’RENCE S
i0
HAYMAN, W.K, er0morphl.c unctios., Oxford, London, 1964.
2.
TSUJI, M, Canonical product for a meromorphfc function in the unit circle,
J. Math. Soc. JaPa_n 8__ (1956), 7-21.
3.
TSUJI, M.
Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959,
SONS, L.R. Value distribution for the derivatives of functions with slow growth
in the disc, preprlnt.
5.
TSUJI, M. Borel’s direction of a meromorphic function fn the unit circle,
J. Math. Soc. Japan 7 (1955), 290-311.
6.
NEVANLINNA, R. Eindeutfge
Berlin, 1953:
analtlsche
Funktfonen, Second Edftlon, Sprlnger-Verlag,
7.
DZHRBASHYAN, M.H.
8.
GIRNYK, M.A. Ein Analogon eines Satzes yon Lindelf 5er den Typ kanonlscher
Produkte, Teor. Funkts. Funkts. Anal. Prilozh. 29 (1978), 16-24,
9.
GIRNYK, M.A. Asymptotic properties of some canonical products II, Siberian
Math. J. 17 (1976), 718-731. Correction Siberian Math. J. 18 (1977), 510.
The theory of factorization and boundary properties of functions
meromorphic in a disc, Russian Math. Surveys 28 (1973), 1-12.
i0. SONS, L.R.
(1982)
Ii. SONS, L.R.
Borel exceptional values in the unit disc, R_ocky Mountain J. Math. 12
197-204.
Value dfstributlon of canonical products in the unit disc,
(1977), 27-38.
ca 22
12. LITTLEWOOD, J.E. Mathematical Notes (II):
J. London Mmth. Soc. 5 (1930), 82-87
Mat__h.
On exceptional values of power series,
13. AHLFORS, L. Ein Satz von Henri Cartan und seine Amendung auf die Theorle der
meromorphen Funktlonen, Soc. Sci. Fenn. Comm. Phys. Math. 16 (1931), 1-19.
14. HAYMAN, W.K. The proximity function in Nevanlinna theory, J. London Math. Soc.
(2), 25 (1982), 473-482.
15. KRUTIN_,
Izl
V.I.
< i,
The size of Nevanlinna
Izv. Akad. Nauk
Arman.
defilencles of functions meromorphlc in
SSSR Ser. Mat. 8__ (1973), 347-358.
16. GIRNYK,
M.A, On the inverse problem of the theory of the distribution of values
for functions that are analytic in the unit disc, Ukrainian Math. J. 29 (1977),
24-30.
17. HAYMAN, W.K. On Nevanlinna’s second theorem and extensions, Rend. Circ. Mat.
Palermo (2) 2 (1953), 1-47.
18. WlTTICH, H. Neuere Untersuchungen uber efndeutlge analytfsche Funktionen,
Zwelte korrigferte Auflage, Sprlnger-Verlag, Berlin, 1968.
19. HAYMAN,
W.K. Pfcard values of meromorphlc functions and their derivatives,
Annals of Math. (2) 70 (1959), 9-42.
20. VALIRON, G,
a_p.
Fonctions analytlques et equations differentlelles,
3_i (19521, 292-303.
J. Math pures et
UNBOUNDED FUNCTIONS IN THE UNIT DISC
239
A general theorem concerning the growth of solutions of first-order
algebraic differential equations, Compositio Math. 25 (1972), 61-70.
21.
BACK, S,
22.
BANK, S.
23.
BANK, S.
24.
GOL’DBERG, A.A.
25.
GOL’DBERG A.A.
26.
MACLNE, G.R.
27.
LEHTO, O. and VIRTANEN, K.I. Boundary behavior and normal meromorphic functions,
Acta Math. 97 (1957), 47-65.
28.
CARTWRIGHT, M.L. and COLLINGWOOD, E.F. Boundary theorem for a function meromorphic in the unit circle, Acta Math. 87 (1952), 83-146.
29.
CARTWRIGHT, M.L. and COLLINGWOOD, E.F. The radial limits of functions meromorphic in a circular disc, Math. Z. 76 (1961), 404-410.
30.
BARTH, K.F. and SCHNEIDER, W.J.
On solutions of algebraic differential equations whose coefficients are
analytic functions in the unit disc, Ann. Mat. Pura. Appl. 92 (1972), 323-335.
On the existence of meromorphic solutions of differential equations
having arbitrarily rapid growth, J. Reine Anew. Mmth. 288 (1976), 176-182.
Growth of functions meromorphic in a disc with restrictions on
the logarithmic derivative, Ukrainian Mmth. J. 32 (1980), 311-316.
On slngle-valued solutions of first order differential equatlons,
Ukrain. Mat. Zh. 8 (1956), 254-261.
Meromorphic functions with small characteristic and no asymptotic
values, Michigan Math. J. 8 (1961), 177-185.
On a problem of Collin.ood concerning meromorphic functions with no asymptotic values, J. London }tth. Soc. (2) i (1969),
553-560.
31.
BARTH, K.F.
Asymptotic values of meromorphlc functions, }ilch. Math. J. 13
(1966),
321-340.
32.
BARTH, K.F. and SCHNEIDER, W.J.
MacLane’s class A
Integrals and derivatives of functions in
and of normal functions, Proc. Camb. Phil. Soc. 71 (1971),
111-121.
33.
BOAS, R.
34.
M_CLANE, G.R. Asymptotic values of holomorphic functions, Rice Univ. Studies
49 (1963), 1-83.
35.
KAPOOR, G.P. On the lower order of functions analytic in the unit disc, Math.
Ja__. 17 (1972), 49-54.
36.
KAPOOR, G.P. and JUNEJA, O.P. On the lower order of functions analytic in the
unit disc II, Indian J. Pure Appl. Math. 7 (1976), 241-248.
37.
SONS, L.R.
296-306.
38.
SHEA, D,F.
39.
LINDEN
40,
LINDEN, C.N. The representation of regular functions, J_, Logdon Math_. Soc.
(1964), 19-30.,
Entire Functions, Academic Press, New York, 1954.
Regularity of growth and gaps, J. Mmth. Anal. App
Corrigendum, J. lNth. Anal. App!. 58 (1977), 232.
!. 2.4
(1968),
Functions analytic in a finite disc and having asymptotically prescribed characteristic, Pacific J. Math. 17 (1966), 549-560.
C.N. Functions analytic in a disc having prescribed asymptotic growth
properties, J. London Math. Soc. (2) 2 (1970), 267272.
3...9
L.R. SONS
240
41.
LINDEN, C.N.
The distribution of the zeros of regular functions, Proc. London
Math. Soc. (3) 15 (1965) 301-322.
42.
BAGEMIHL. F.; ERDOS, P. and SEIDEL, W, Sur quelques propri4t4s frontiefes
des fonctions holomorphes d4flnes par certalns prodults dans le cercleun+/-e, An_n_. _S.ci. .coe._o.rm._Sup. (3) 70 (1953), 135-3.47.
43.
MACLANE, G.R.
44.
BAGEMIHL, F.
45.
HEINS, M.
Holomorphic functions, of arbitrarily slow growth, without radial
limits, Mich. }th. J, 9 (1962), 21-24.
Curvillnear cluster sets of arbitrary functions,
Sci. U.S,A. 41 (1955) 379382.
_Pro___qc. Na___t. Acad.
A property of the asymptotic spots of a meromorphic function or an
interior transformation whose domain is the open unit disk, J. Indian Mmth.
Soc. 24 (1960), 265-268.
A,
46.
HORNBLOWER, R. A growth condition for the MacLane class
Soc. (3) 23 (1971), 371-384.
47.
MACLANE, G.R. A growth condition for class A, _Mich. Math.
48.
RIPPON, P.J. On a growth condition related to the cLane class, J. London Math.
Soc (2) 18 (1978), 94-i00.
49.
GROSZ, W. Eine ganze Funktion, fur die jede komplexe Zahl Konvergenzwert ist,
Math. Ann. 79 (1918), 201-208.
50.
DRASIN, D.
51.
MCLLAN, J.E.
2_5 (1978), 263-287.
Some problems related to MacLane’s class, J. London l-th. Soc. (2)
23 (1981), 280-286.
Asymptotic values of functions holomorphic in the unit disc,
Mich. Math. J.
52.
LINDEN, C.N.
53.
LINDEN, C.N.
1__2 (1965), 141-154.
The minimum modulus of functions regular and of finite order in
the unit circle, Quart. J. Math. Oxford (2) 7 (1956), 196-216.
_[nlmum
Quart. J. Math.
54.
J.
Proc. London Math.
LINDEN, C.N.
Mathematical
modulus theorems for functions regular in the unit circle,
Oxfor.d
(2) 1__2 (1961), 1-16.
"The minimum modulus of functions of slow growth in the unit disk,’:
Essays Dedicated to A.J. Mmclntvre, Ohio University Press, Athens
(Ohio), 19 70.
55.
ARIMA, K.
On maximum modulus of integral functions, J. Math. Soc. Japan
4_ (1952),
62-66.
56.
BELLER, E.
57.
p
BELLER, E. Zeros of A functions and related classes of’analytlc functions,
Israel J. Math. 22 (1975), 68-80.
58.
HEILPER, A,
Factorlzatlon for non-Nevanllnna classes of analytic functions,
Israel J. Math. 27 (1977), 320-330.
The zeros of functions in NevanlinnaVs area class, Israel J. Math.
34 (1980), i-ii.
59.
SHAPIRO, H,S. and SHIELDS, A,L,
60.
KAYMAN, W,K. and KORENBLUM, B. A critical growth rate for functions regular in
a disk, Mich. Math. J. 27 (1980), 21-30.
On the zeros of functions with finite Dirichlet
integral and some related function spaces, Math, Z. 80 (1962), 217-229.
241
UNBOUNDED FUNCTIONS IN THE UNIT DISC
61.
LINDEN, C,N,
Regular functions of restricted growth and their zeros in tangen-
tial regions, preprint.
62.
HANSON, B.
63,
MURAl, T.
Ph,D. Dissertation, University of Wisconsin, preprint,
The value distribution of lacunary series and a conjecture of Paley,
Ann. Inst. Fourier (Grenoble) 31 (1981), 136-156.
64.
}fl/RAI, T, Sur la distribution des valeurs des sdries lacunaires, J. London Math.
Soc. (2) 21 (1980), 93-110.
65.
MURAl, T.
66.
WIMAN, A.
67.
SONS, L.R.
25-48.
68.
NICHOLLS, P.J. and SONS, L.R.
69.
SONS, L.R. Value distribution and power series with moderate gaps, }’./ch. Math.
J. 13 (1966), 425-433.
70.
SONS, L.R.
71.
NEVANLINNA, R.
The boundary behavior of Hadamard lacunary series, Nagoya
89 (1983), to appear.
th.o
J.
Uber den Zusammenhang zwischen dem Maximalbetrage einer analytischen
Funktion und dem grssten Betrage bei gegebenen Argumente der Funktlon, Acta.
Math. 41 (1916), 1-23.
Gap series with rapid growth in the unit disc, Math. Z.
13i (1973),
Minimum modulus and zeros of functions in the unit
disc, Proc. London Math. Soc. (3) 31 (19 75), 99-113.
Asymptotic paths for power serles with exponents of the form n
Mich. Math. J. 19 (1972), 383-384.
h,
p,
Le Theoreme de Picard-Borel et la theorie des fonctions meromorGauthier-Villars, Paris, 1929.
72.
HAYMAN, W.K. and STORVICK, D.A.
3 (1971), .193-194.
73.
SONS, L.R. Gap series
7 (1981), 431-440.
74.
TSUJI, M.
75.
On the characteristic of functions merDmorphic in the unit disk
HAYMAI], U.K.
and of their integrals, Acta Math. 112 (1964), 181-214.
76.
DUREN, P.
77.
HAYMAN, W.K.
78.
POMMERENKE, C.
79.
CLUNIE, J. On integral functions having prescribed asymptotic growth, Canadian
J. Math. 17 (1965), 396-404.
80.
HOROWITZ, C.
693-710.
81.
KORENBLUM, B,
187-219.
82.
SONS, L.R. Zeros of power
(1972), 199-205.
On normal functions, Bull. London Math. Soc.
growth, normality, and values assumed, Houston J. Math.
Theory of Fuchsian groups, Ja2. J. th. 2__1 (1951), 1-27.
_The?ry
of
Hp
Spa.ces Academic Press, New York and London, 1970.
Multivalent Functions, Cambridge Univ. Press, London, 1967.
Univalent Functions, Vandenhoeck and Ruprecht, Gttlngen, 1975.
Zeros of functions in the Bergman spaces, Duke Math:,;J.,
An extension of the Nevanlinna theory,
series with large gaps,
Acta Math_.
Indiana U.niv._
4_I (1974),
135 (1975),
Math, J.
2_2
L.R. SONS
242
83.
NDERSON,.J,M.;
functions,
J,
J, and POMRENKE, C. On Bloch functions and normal
ineangewndte Mat_h, 2/__0 (1974), 12-37,
CLUNIE
The distribution of the values of a random unction in the unit
disk, Studia Math, 41 (1972), 71I06,
84.
OFFORD, A.C.
85.
BAGEMIHL, F. and
holomorphic
ERDS,
(1965), 340-344.
86.
BONAR, D.D0
P.
A problem concerning the zeros of a certain kind of
function in the unit
circle J.
reine
anewandte
Math.
214215
On Annular Functions, VEB Deutscher Verlag der Wissenschaften,
Berlin, 19 71.
87.
BONAR, D, and CARROLL, F. Fatou points for annular functions, J._ reine anKewandte__
Ma__, 283/-284 (1976), 222226.
88.
SONS, L,R. Zeros of functions with slow growth in the
24 (1979), 271-282.
89.
SONS, L.R.
unit disc, Math. Japon.i_ca
Zero distribution of functions with slow on moderate growth in the
unit disc, Pacific J. Mmth. 99 (1982), 473-481.