- Research Article
- Open Access
Normality of Composite Analytic Functions and Sharing an Analytic Function
© Qifeng Wu et al. 2010
- Received: 17 August 2010
- Accepted: 15 October 2010
- Published: 18 October 2010
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 .
- Analytic Function
- Composite Analytic
- Computational Biology
- Accumulation Point
- Normal Family
In 1952, Rosenbloom  proved the following theorem.
Influenced from Bloch's principle ( or ), that is, there is a normal criterion corresponding to every Liouville-Picard type theorem, Fang and Yuan  proved a corresponding normality criterion for inequality (1.2).
In 1995, Zheng and Yang  proved the following result.
In 2000, Fang and Yuan  improved (1.3) and obtained the best possible .
The corresponding normal criterion below to Theorem D was obtained by Fang and Yuan .
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  proved the following theorem.
In 2004, Bergweiler  deals also with the case that is meromorphic in Theorem F and extended Theorem E as follows.
Recently, Yuan et al.  generalized Theorem G in another manner and proved the following result.
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:
(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 .
In this paper, we improve Theorems E and F and obtain the main result Theorem 1.1 which is proved below in Section 3.
In order to prove our result, we need the following lemmas. Lemma 2.1 is an extending result of Zalcman  concerning normal families.
Lemma 2.1 (see ).
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.
If is a transcendental entire function, then either or has infinite zeros. Indeed, suppose that it is not true, then by Picard's theorem , we obtain that is a polynomial, a contradiction.
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.
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 .
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.
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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