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Normality of Composite Analytic Functions and Sharing an Analytic Function
Fixed Point Theory and Applications volume 2010, Article number: 417480 (2010)
Abstract
A result of Hinchliffe (2003) is extended to transcendental entire function, and an alternative proof is given in this paper. Our main result is as follows: let 
be an analytic function, 
a family of analytic functions in a domain 
, and 
a transcendental entire function. If 
and 
share 
IM for each pair 
, and one of the following conditions holds: (1) 
has at least two distinct zeros for any 
; (2) 
is nonconstant, and there exists 
such that 
has only one distinct zero 
, and suppose that the multiplicities 
and 
of zeros of 
and 
at 
, respectively, satisfy 
, for each 
, where 
; (3) there exists a 
such that 
has no zero, and 
is nonconstant, then 
is normal in 
.
1. Introduction and Main Results
Let 
and 
be two nonconstant meromorphic functions in the whole complex plane 
, and let 
be a finite complex value or function. We say that 
and 
share 
CM (or IM) provided that 
and 
have the same zeros counting (or ignoring) multiplicity. It is assumed that the reader is familiar with the standard notations and the basic results of Nevanlinna's value-distribution theory
([1] or [2]). We denote by 
any function satisfying 
as 
, possibly outside of a set of finite measure.
A meromorphic function 
is called a small function related to 
if 
.
In 1952, Rosenbloom [3] proved the following theorem.
Theorem A.
Let 
be a polynomial of degree at least 
and 
a transcendental entire function. Then
Influenced from Bloch's principle ([1] or [4]), that is, there is a normal criterion corresponding to every Liouville-Picard type theorem, Fang and Yuan [5] proved a corresponding normality criterion for inequality (1.2).
Theorem B.
Let 
be a family of analytic functions in a domain 
and 
a polynomial of degree at least 
. If 
for each 
, then 
is normal in 
.
In 1995, Zheng and Yang [6] proved the following result.
Theorem C.
Let 
be a polynomial of degree 
at least 
, 
a transcendental entire function, and 
a nonconstant meromorphic function satisfying 
. Then,
Here 
if 
has only one zero; otherwise 
.
In 2000, Fang and Yuan [7] improved (1.3) and obtained the best possible 
.
Theorem D.
Let 
be a polynomial of degree 
at least 
and 
a transcendental entire function, and 
a nonconstant meromorphic function satisfying 
. If 
is a constant, we also require that there exists a constant 
such that 
has a zero of multiplicity at least 2. Then
Here 
if 
has only one zero; otherwise 
.
The corresponding normal criterion below to Theorem D was obtained by Fang and Yuan [7].
Theorem E.
Let 
be a family of analytic functions in a domain 
and 
a polynomial of degree at least 
. Suppose that 
is either a nonconstant analytic function or a constant function such that 
has at least two distinct zeros. If 
for each 
, then 
is normal in 
.
In 2003, Hinchliffe [8] proved the following theorem.
Theorem F.
Let 
, 
a family of analytic functions in a domain 
, and 
a transcendental meromorphic function. If 
, 
or 
, where 
are two distinct values in 
, suppose that 
for each 
and all 
. Then, 
is normal in 
In 2004, Bergweiler [9] deals also with the case that 
is meromorphic in Theorem F and extended Theorem E as follows.
Theorem G.
Let 
be a nonconstant meromorphic function, 
a family of analytic functions in a domain 
, and 
a rational function of degree at least 2. Suppose that 
for each 
and all 
. Then, 
is normal in 
Recently, Yuan et al. [10] generalized Theorem G in another manner and proved the following result.
Theorem H.
Let 
be a nonconstant meromorphic function, 
a family of analytic functions in a domain 
, and 
a rational function of degree at least 
. If 
and 
share 
IM for each pair 
, 
and one of the following conditions holds:
(1)
has at least two distinct zeros or poles for any 
;
(2)there exists 
such that 
has only one distinct zero (or pole) 
and suppose that the multiplicities 
and 
of zeros of 
and 
at 
, respectively, satisfy 
(or 
), for each 
, where 
and 
are two of no common zero polynomials with degree 
and 
, respectively, and 
.
Then, 
is normal in 
.
In this paper, we improve Theorems E and F and obtain the main result Theorem 1.1 which is proved below in Section 3.
Theorem 1.1.
Let 
be an analytic function, 
a family of analytic functions in a domain 
, and 
a transcendental entire function. If 
and 
share 
IM for each pair 
, and one of the following conditions holds:
(1)
has at least two distinct zeros for any 
;
(2)
is nonconstant, and there exists 
such that 
has only one distinct zero 
and suppose that the multiplicities 
and 
of zeros of 
and 
at 
, respectively, satisfy 
, for each 
, where 
;
(3)there exists a 
such that 
has no zero, and 
is nonconstant.
Then, 
is normal in 
.
2. Preliminary Lemmas
In order to prove our result, we need the following lemmas. Lemma 2.1 is an extending result of Zalcman [11] concerning normal families.
Lemma 2.1 (see [12]).
Let 
be a family of functions on the unit disc. Then, 
is not normal on the unit disc if and only if there exist
(a)a number 
(b)points 
with 
(c)functions 
;
(d)positive numbers 
such that 
converges locally uniformly to a nonconstant meromorphic function 
, which order is at most 2.
Remark 2.2.
is a nonconstant entire function if 
is a family of analytic functions on the unit disc in Lemma 2.1.
The following Lemma 2.3 is very useful in the proof of our main theorem. We denote by 
the open disc of radius 
around 
, that is, 
. 
Let 
be a family of analytic functions in 
. Suppose that 
is not normal at 
but is normal in 
. Then, there exists a subsequence 
of 
and a sequence of points 
tending to 
such that 
, but 
tending to infinity locally uniformly on 
.
3. Proof of Theorem
Proof of Theorem 1.1.
Without loss of generality, we assume that 
. Then, we consider three cases:
Case 1.
has at least two distinct zeros for any 
Suppose that 
is not normal in 
. Without loss of generality, we assume that 
is not normal at 
.
Set 
have two distinct zeros 
and 
.
By Lemma 2.1, there exists a sequence of points 
, 
and 
such that
uniformly on any compact subset of 
, where 
is a nonconstant entire function.
Hence,
uniformly on any compact subset of 
.
We claim that 
had at least two distinct zeros.
If 
is a nonconstant polynomial, then both 
and 
have zeros. So 
has at least two distinct zeros.
If 
is a transcendental entire function, then either 
or 
has infinite zeros. Indeed, suppose that it is not true, then by Picard's theorem [2], we obtain that 
is a polynomial, a contradiction.
Thus, the claim gives that there exist 
and 
such that
We choose a positive number 
small enough such that 
and 
has no other zeros in 
except for 
and 
, where
By hypothesis and Hurwitz's theorem [14], for sufficiently large 
there exist points 
, 
such that
Note that 
and 
share 
IM; it follows that
Taking 
, we obtain
Since the zeros of
have no accumulation points, we have
or equivalently
This contradicts with the facts that 
, 
, and 
.
Case 2.
is nonconstant, and there exists 
such that 
has only one distinct zero 
, and suppose that the multiplicities 
and 
of zeros of 
and 
at 
, respectively, satisfy 
, possibly outside finite 
, where 
.
We shall prove that 
is normal at 
. Without loss of generality, we can assume that 
.
By 
nonconstant and analytic, we see that there exists a neighborhood 
such that
Hypothesis implies that 
has only one zero 
, that is, 
.
We claim that 
is normal at 
for small enough 
. In fact, 
has infinite zeros by Picard theorem. Hence, the conclusion of Case 1 tells us that this claim is true.
Next, we prove 
is normal at 
. For any 
, by the former claim, there exists a subsequence of 
, denoted 
for the sake of simplicity, such that
uniformly on a punctured disc 
.
By hypothesis, we see that 
is an analytic family in the disc 
.
If 
is not normal at 
, then Lemma 2.3 gives that 
on a punctured disc 
and 
for a sequence of points 
.
We claim that there exists a sequence of points 
(
) such that 
.
In fact we may find 
such that 
for 
Next, we choose 
with 
such that 
for 
Since 
on 
and 
for a sequence of points 
, we know that if 
sufficiently large, then
for 
and 
For large 
, we also have 
, and thus we deduce that from Rouché's theorem that 
takes the value 
, that is, we have 
for large 
. Since also 
for large 
, we find a component 
of 
contained in 
for such 
. Moreover, 
is a Jordan domain, and 
is a proper map.
For 
, we then have 
, and thus 
. Hence
for 
. Now 
, in particular, takes the value 
in 
, say, 
with 
Hence, 
, and thus Rouché's theorem now shows that our claim holds.
By the similar argument as Case 1, we obtain that 
for sufficiently large 
. Because 
, we have
Hence,
where 
, 
are analytic functions and 
, 
Set 
, if 
or 
if 
. Thus, 
or 
. Noting that 
, we see that 
is an analytic family and normal in 
.
By the same argument as above, there exists a sequence of points 
such that 
, and 
. Obviously, 
and
Noting that 
and 
share 
IM, we obtain that
for each 
. That is, 
. Noting that 
, we deduce that 
. Thus, taking 
, 
contradicting the hypothesis for 
.
Case 3.
There exists a 
such that 
has no zero, and 
is nonconstant.
Suppose that 
is not normal in 
. Without loss of generality, we assume that 
is not normal at 
.
By Picard theorem and (3.11), we know that 
has at least two distinct zeros at any 
for small enough 
. The result of Case 1 tell us that 
is normal in 
.
Thus, for any 
, by the former conclusion and Lemma 2.3, there exists a subsequence of 
, denoted by 
for the sake of simplicity, such that
uniformly on a punctured disc 
and 
for a sequence of points 
.
Obviously, 
is an analytic normal family in the punctured disc 
for small enough 
. We consider two subcases.
Subcase 1 (
is not normal at 
).
Using Lemma 2.3 for 
, we get that there exists a sequence of points 
such that 
and 
.
Noting that 
and 
share 
IM, and 
has no zero, it follows that 
and 
Taking 
, we obtain 
A contradiction with the hypothesis that 
has no zero.
Subcase 2 (
is normal at 
).
Then, 
is normal in 
, which tends to a limit function 
, which is either identically infinite or analytic in 
. Set
noting that 
as 
. If 
is large enough, we have 
, and hence 
. Denote 
by 
, and note that the 
are closed curves, arbitrarily distant from and surrounding the origin.
Suppose that 
on 
. Since 
locally uniformly on 
, there exists, for arbitrarily large positive 
, an 
such that, for 
, 
on 
. Thus, we have 
on 
. Hence, for large 
is bounded away from 
on the curves 
, and this contradicts Iversen's theorem [15].
On the other hand, suppose that 
is analytic on 
. Then, there exists some constant 
such that 
on 
, and so, for large 
on 
. Hence, 
on 
. Again, 
is therefore bounded away from 
of its omitted value on the curves 
, contradicting Iversen's theorem.
Therefore 
is normal in Case 3.
Theorem 1.1 is proved completely.
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Acknowledgments
The authors would like to express their hearty thanks to Professor Mingliang Fang and Degui Yang for their helpful discussions and suggestions. The authors would like to thank referee for his (or her) very careful comments and helpful suggestions. This paper is supported by the NSF of China (no. 10771220), Doctorial Point Fund of National Education Ministry of China (no. 200810780002), and Guangzhou Education Bureau (no. 62035).
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Wu, Q., Xiao, B. & Yuan, W. Normality of Composite Analytic Functions and Sharing an Analytic Function. Fixed Point Theory Appl 2010, 417480 (2010). https://doi.org/10.1155/2010/417480
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DOI: https://doi.org/10.1155/2010/417480