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On Entire and Meromorphic Functions That Share One Small Function with Their Differential Polynomial

DOI: 10.1155/2013/926340

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We study the uniqueness of meromorphic functions that share one small function with more general differential polynomial . As corollaries, we obtain results which answer open questions posed by Yu (2003). 1. Introduction and Main Results In this paper, a meromorphic functions mean meromorphic in the whole complex plane. We use the standard notations of Nevanlinna theory (see [1]). A meromorphic function is called a small function with respect to if , that is, as possibly outside a set of finite linear measure. If and have the same zeros with same multiplicities (ignoring multiplicities), then we say that and share ?CM (IM). For any constant , we denote by the counting function for zeros of with multiplicity no more than and the corresponding for which multiplicity is not counted. Let be the counting function for zeros of with multiplicity at least and the corresponding for which the multiplicity is not counted. Let and be two nonconstant meromorphic functions sharing value 1?IM. Let be common one point of and with multiplicity and , respectively. We denote by ?? the counting (reduced) function of those 1 points of where ; by the counting function of those 1-points of where ; by the counting function of those 1-points of where . In the same way, we can define , and (see [2]). In 1996, Brück [3] posed the following conjecture. Conjecture 1. Let be a nonconstant entire function such that the hyper-order of is not a positive integer and . If and share a finite value ?CM, then , where is a nonzero constant. In [3], under an additional hypothesis, Brück proved that the conjecture holds when . Theorem A. Let be a nonconstant entire function. If and share the value 1?CM and if , then , for some constant . Many people extended this theorem and obtained many results. In 2003, Yu [4] proved the following theorem. Theorem B. Let . Let be a nonconstant meromorphic function and a meromorphic function such that , and do not have any common pole and as . If and share the value 0?CM and then . Theorem C. Let . Let be a nonconstant entire function and be a meromorphic function such that and as . If and share the value 0?CM and then . In the same paper, the author posed the following questions. Question 1. Can ?CM shared value be replaced by an IM shared value in Theorem C? Question 2. Is the condition sharp in Theorem C? Question 3. Is the condition sharp in Theorem B? In 2004, Liu and Gu [5] applied different method and obtained the following theorem which answers some questions posed in [4]. Theorem D. Let . Let be a nonconstant meromorphic function and a meromorphic

References

[1]  H. X. Yi and C. C. Yang, Uniqueness Theory of Meromorphic Functions, Science Press, Beijing, China, 1995.
[2]  W.-C. Lin and H.-X. Yi, “Uniqueness theorems for meromorphic function,” Indian Journal of Pure and Applied Mathematics, vol. 35, no. 2, pp. 121–132, 2004.
[3]  R. Brück, “On entire functions which share one value CM with their first derivative,” Results in Mathematics, vol. 30, no. 1-2, pp. 21–24, 1996.
[4]  K.-W. Yu, “On entire and meromorphic functions that share small functions with their derivatives,” Journal of Inequalities in Pure and Applied Mathematics, vol. 4, no. 1, article 21, 2003.
[5]  L. Liu and Y. Gu, “Uniqueness of meromorphic functions that share one small function with their derivatives,” Kodai Mathematical Journal, vol. 27, no. 3, pp. 272–279, 2004.
[6]  T. Zhang and W. Lü, “Notes on a meromorphic function sharing one small function with its derivative,” Complex Variables and Elliptic Equations, vol. 53, no. 9, pp. 857–867, 2008.
[7]  S. S. Bhoosnurmath and A. J. Patil, “On the growth and value distribution of meromorphic functions and their differential polynomials,” The Journal of the Indian Mathematical Society, vol. 74, no. 3-4, pp. 167–184, 2007.
[8]  H. X. Yi, “Uniqueness of meromorphic functions and a question of C. C. Yang,” Complex Variables—Theory and Application, vol. 14, no. 1–4, pp. 169–176, 1990.
[9]  H.-X. Yi, “Uniqueness theorems for meromorphic functions whose n-th derivatives share the same 1-points,” Complex Variables—Theory and Application, vol. 34, no. 4, pp. 421–436, 1997.
[10]  N. Li and L.-Z. Yang, “Meromorphic function that shares one small function with its differential polynomial,” Kyungpook Mathematical Journal, vol. 50, no. 3, pp. 447–454, 2010.

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