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 .
1. Introduction and Main Results
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.
2. Preliminary Lemmas
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 .
3. Proof of Theorem
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).
- Hayman WK: Meromorphic Functions, Oxford Mathematical Monographs. Clarendon Press, Oxford, UK; 1964:xiv+191.Google Scholar
- Yang L: Value Distribution Theory. Springer, Berlin, Germany; 1993:xii+269.MATHGoogle Scholar
- Rosenbloom PC: The fix-points of entire functions. Comm. Sem. Math. Univ. Lund, Tom Supplementaire 1952, 1952: 186–192.MathSciNetGoogle Scholar
- Bergweiler W: Bloch's principle. Computational Methods and Function Theory 2006,6(1):77–108.MathSciNetView ArticleMATHGoogle Scholar
- Fang ML, Yuan W: On the normality for families of meromorphic functions. Indian Journal of Mathematics 2001,43(3):341–351.MathSciNetMATHGoogle Scholar
- Zheng JH, Yang C-C: Further results on fixpoints and zeros of entire functions. Transactions of the American Mathematical Society 1995,347(1):37–50. 10.2307/2154786MathSciNetView ArticleMATHGoogle Scholar
- Fang ML, Yuan W: On Rosenbloom's fixed-point theorem and related results. Australian Mathematical Society Journal Series A 2000,68(3):321–333. 10.1017/S1446788700001415MathSciNetView ArticleMATHGoogle Scholar
- Hinchliffe JD: Normality and fixpoints of analytic functions. Proceedings of the Royal Society of Edinburgh Section A 2003,133(6):1335–1339. 10.1017/S0308210500002961MathSciNetView ArticleMATHGoogle Scholar
- Bergweiler W: Fixed points of composite meromorphic functions and normal families. Proceedings of the Royal Society of Edinburgh. Section A 2004,134(4):653–660. 10.1017/S0308210500003401MathSciNetView ArticleMATHGoogle Scholar
- Yuan WJ, Li ZR, Xiao B: Normality of composite analytic functions and sharing a meromorphic function. Science in China Series A 2010,40(5):429–436.Google Scholar
- Zalcman L: A heuristic principle in complex function theory. The American Mathematical Monthly 1975,82(8):813–817. 10.2307/2319796MathSciNetView ArticleMATHGoogle Scholar
- Zalcman L: Normal families: new perspectives. American Mathematical Society Bulletin. New Series 1998,35(3):215–230. 10.1090/S0273-0979-98-00755-1MathSciNetView ArticleMATHGoogle Scholar
- Clifford EF: Normal families and value distribution in connection with composite functions. Journal of Mathematical Analysis and Applications 2005,312(1):195–204. 10.1016/j.jmaa.2005.03.045MathSciNetView ArticleMATHGoogle Scholar
- Gu YX, Pang XC, Fang ML: Theory of Normal Family and Its Applications. Science Press, Beijing, China; 2007.Google Scholar
- Tsuji M: Potential Theory in Modern Function Theory. Maruzen, Tokyo, Japan; 1959:590.MATHGoogle Scholar
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