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International Journal of Analysis
Volume 2013 (2013), Article ID 451468, 4 pages
http://dx.doi.org/10.1155/2013/451468
Research Article

Normal Families concerning Polynomials and Shared Values

1College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
2School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Received 8 July 2013; Revised 14 September 2013; Accepted 15 September 2013

Academic Editor: Wen-Xiu Ma

Copyright © 2013 Xin-Li Wang and Ning Cui. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the problem of normal families of meromorphic functions concerning polynomials and shared values. We prove that a family of meromorphic functions in a domain D is normal if, for each function , , where is a polynomial with the origin as zero, is a positive integer, and , are two finite constants.

1. Introduction and Main Results

Let D be a domain in and let be a family of meromorphic functions defined in D. is said to be normal in D, in the sense of Montel, if, for any sequence , there exists a subsequence such that converges spherically locally uniformly in D to a meromorphic function or (see [1]).

Let and be meromorphic functions in D, and let and be complex numbers. If whenever , we write If and , we write If , we say that and share on D.

In 1992, Schwick [2] firstly found a connection between normality criteria and shared values as follows.

Theorem A. Let be a family of meromorphic functions in D, and let be distinct complex numbers. If, for each , and share , then is normal in D.

In recent years, more results about normality criteria concerning shared values have been found (see [35]). The next theorem was proved by Fang and Zalcman in [4], which was an important normality criteria concerning shared values.

Theorem B. Let be a family of meromorphic functions in D; let be a positive integer; let and be two finite values. If, for every , all zeros of f have multiplicity at least and , then is normal in D.

Based on the idea in [3], we will improve and extend Theorem B in the following results.

Theorem 1. Let be a family of meromorphic functions in D; let be a positive integer; let and be two finite values. Let P be a polynomial with the origin as zero. If, for every , all zeros of f have multiplicity at least and , then is normal in D.

Example 2. Let , , . Then, all zeros of have multiplicity at least 2. For each , we have , . Thus, . , but is not normal in D. This shows that the condition is necessary.

Theorem 3. Let be a family of meromorphic functions in D; let be a positive integer; let and be two finite values. Let P be a polynomial with the origin as zero and , which has at least one multiple zero. If, for every , all zeros of have multiplicity at least and , then is normal in D.

2. Some Lemmas

In order to prove our results, we need the following lemmas.

Lemma 1 (see [6]). Let be a family of meromorphic functions on the unit disc satisfying all zeros of functions in which have multiplicity ≥p and all poles of functions in . Let be a real number satisfying . Then, is not normal at a point if and only if there exist(i)points , ;(ii)positive numbers , ;(iii)functions such that spherically uniformly on each compact subset of , where is a nonconstant meromorphic function satisfying the zeros of which are of multiplicities and the poles of which are of multiplicities . Moreover, the order of is not greater than 2.

Here, is the spherical derivative.

Lemma 2 (see [7]). Let be a family of meromorphic functions on the complex plane . If the spherical derivative of is bounded on , then the order of is at most two. Especially, when is an entire function on the complex plane , then the order of is at most one.

Lemma 3 (see [8]). Let be a transcendental meromorphic function on the complex plane , and let be a positive integer. Then assumes every nonzero finite complex value infinitely often.

Lemma 4 (see [9]). Let and be two positive integers, and let be a transcendental meromorphic function. Then assumes every nonzero finite complex value infinitely often.

Lemma 5 (see [5]). Let be a nonzero finite complex number, let be two positive integers, and let be a nonconstant rational function; all of whose zeros have multiplicities at least . For the polynomial case, if or , then has at least two distinct zeros; for the nonpolynomial rational case, the conclusion holds if .

3. Proof of Theorems

We only prove Theorem 1, and the proof of Theorem 3 is similar to the second case in the proof of Theorem 1.

Next, we consider two cases.

Case 1. Suppose the origin is a simple zero of . We assume that , where . Without loss of generality, we may assume that . Since normality is a local property, it is enough to show that is normal at each point of D. Suppose, on the contrary, that is not normal at . By Lemma 1 (with ), there exist functions , points , and positive numbers such that converges spherically uniformly on compact subsets of , where is a nonconstant meromorphic function on and all of whose zeros have multiplicity at least . Moreover, the order of is at most 2. Therefore, we have uniformly on compact subsets of disjoint from the poles of .
We claim (i) ; (ii) .(i) Suppose that , then . If , is an entire function and . By Lemma 2, the order of is at most 1. Assume , where and are two constants. But , which is a contradiction. Therefore, we have . According to Hurwitz’s theorem, there exists a sequence , such that for sufficiently large. It follows that , so that Thus which contradicts . This proves (i).(ii) Suppose that , then . If , is a polynomial whose degree is less than , which contradicts that all zeros of have multiplicity at least . So that . Since , by Hurwitz’s theorem, there exists a sequence , such that for sufficiently large. It follows that , so that which contradicts . This proves (ii).
Next, we assume that is a transcendental meromorphic function. When , by Lemma 3, assumes every nonzero finite complex value infinitely often, a contradiction. When , by Nevanlinna's second fundamental theorem, where .
By Nevanlinna’s first fundamental theorem, we have Combining (8) with (9), we have According to logarithmic derivative theorem, we have It follows that Since and the zeros of have multiplicity exactly , we have by (10) and (13) Thus , a contradiction. Hence is a rational function.
Now, suppose that is a nonpolynomial rational function. Let where are constants such that and and are coprime polynomials with and . If , where and are coprime polynomials and, by induction, . Then, by (16), has solutions, a contradiction. Hence . Thus, we have Since , is a nonzero constant . Since , must have a pole in the finite plane. It follows that has a finite pole of order at least ; thus, , Then, the equation has solutions, which contradict .
Finally, suppose that is a polynomial. Since and the zeros of are of multiplicity , we have , where . Thus, has solutions, a contradiction.

Case 2. Suppose the origin is a multiple zero of . Without loss of generality, we assume that . Let , where is a positive integer and are constants. Suppose is not normal at . By Lemma 1 (with ), there exist functions , points , and positive numbers such that converges spherically uniformly on compact subsets of , where is a nonconstant meromorphic function on , and all of whose zeros have multiplicity at least . Moreover, the order of is at most 2. We have uniformly on compact subsets of disjoint from the poles of . If , has no zeros and poles. That is to say, is an entire function and . Since the spherical derivative of is bounded, by Lemma 2, we have . Hence suppose , where and are constants. Thus, , a contradiction. By Lemmas 4 and 5, has zeros. Suppose . Since , by Hurwitz’s theorem, there exists a sequence , such that (for sufficiently large) It follows that , so that Thus, which contradicts . This completes the proof.

Acknowledgment

This paper is supported by the National Natural Science Foundation of China (no. 61074016).

References

  1. W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, UK, 1964. View at MathSciNet
  2. W. Schwick, “Sharing values and normality,” Archiv der Mathematik, vol. 59, no. 1, pp. 50–54, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. C. L. Lei and M. L. Fang, “Normality and shared values concerning differential polynomials,” Science China, vol. 53, no. 3, pp. 749–754, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. L. Fang and L. Zalcman, “Normal families and shared values of meromorphic functions II,” Computational Methods and Function Theory, vol. 1, no. 1, pp. 289–299, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. X. Wu and Y. Xu, “Normal families of meromorphic functions and shared values,” Monatshefte für Mathematik, vol. 165, no. 3-4, pp. 569–578, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. L. Zalcman, “Normal families: new perspectives,” Bulletin of the American Mathematical Society, vol. 35, no. 3, pp. 215–230, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. J. Clunie and W. K. Hayman, “The spherical derivative of integral and meromorphic functions,” Commentarii Mathematici Helvetici, vol. 40, pp. 117–148, 1966. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. W. Bergweiler and A. Eremenko, “On the singularities of the inverse to a meromorphic function of finite order,” Revista Matemática Iberoamericana, vol. 11, no. 2, pp. 355–373, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. N. Steinmetz, “Über die nullstellen von differentialpolynomen,” Mathematische Zeitschrift, vol. 176, no. 2, pp. 255–264, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet