Normality of Composite Analytic Functions and Sharing an Analytic Function

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 .

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 http://D webcite 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 http://F webcite and extended Theorem http://E webcite 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 http://G webcite 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 http://E webcite and http://F webcite 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 .

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, .

Lemma 2.3 (see [13] or [14]).

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 .

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.

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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