Value Sharing of Certain Differential Polynomials and Their Shifts of Meromorphic Functions
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Value Sharing of Certain Differential Polynomials and Their Shifts of Meromorphic Functions Xiao-Min Li · Hong-Xun Yi · Yue Shi
Received: 18 July 2013 / Revised: 9 December 2013 / Accepted: 11 December 2013 / Published online: 5 March 2014 © Springer-Verlag Berlin Heidelberg 2014
Abstract Using Zalcman’s lemma, we study a uniqueness question of meromorphic functions, where the meromorphic functions and their certain non-linear differential polynomials share a non-zero value with their shifts. The results in this paper extends Theorem 1 in Yang and Hua (Ann Acad Sci Fenn Math 22:395–406, 1997) and Theorem 1 in Fang (Comput Math Appl 44:823–831, 2002). Our reasoning in this paper also corrects the proof of Theorem 4 in Bhoosnurmath and Dyavanal (Comput Math Appl 53:1191–1205, 2007). Keywords Meromorphic functions · Shared values · Differential polynomials · Uniqueness theorems
Communicated by Ilpo Laine. This work is supported by the NSFC (No. 11171184), the NSFC and RFBR (Joint Project) (No. 10911120056), the NSF of Shandong Province, China (No. Z2008A01), and the NSF of Shandong Province, China (No. ZR2009AM008). X.-M. Li (B)· Y. Shi Department of Mathematics, Ocean University of China, Qingdao 266100, Shandong, People’s Republic of China e-mail: [email protected] Y. Shi e-mail: [email protected] X.-M. Li Department of Physics and Mathematics, University of Eastern Finland, P. O. Box 111, 80101 Joensuu, Finland H.-X. Yi Department of Mathematics, Shandong University, Jinan 250100, Shandong, People’s Republic of China e-mail: [email protected]
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Mathematics Subject Classification (2010)
30D35 · 30D30
1 Introduction and Main Results In this paper, by meromorphic functions we will always mean meromorphic functions in the complex plane. We adopt the standard notation in the Nevanlinna theory of meromorphic functions as explained in [9,14,23,24]. It will be convenient to let E denote any set of positive real numbers of finite linear measure, not necessarily the same at each occurrence. For a non-constant meromorphic function h, we denote by T (r, h) the Nevanlinna characteristic of h and by S(r, h) any quantity satisfying S(r, h) = o{T (r, h)}, as r → ∞ and r ∈ E. Let f and g be two non-constant meromorphic functions, and let a be a finite complex number. We say that f and g share a CM, provided that f − a and g − a have the same zeros with the same multiplicities. Similarly, we say that f and g share a IM, provided that f − a and g − a have the same zeros ignoring multiplicities. In addition, we say that f and g share ∞ CM, if 1/ f and 1/g share 0 CM, and we say that f and g share ∞ IM, if 1/ f and 1/g share 0 IM (see [23]). We say that a is a small function of f, if a is a meromorphic function satisfying T (r, a) = S(r, f ) (see[23]). Throughout this paper, we denote by μ( f ), ρ( f ), ρ2 ( f ) and λ( f ) the lower order of f, the order of f, the hyper-order of f and the exponent of convergence of zeros of f respectively (see [9,14,23,24]). In addition, we need the following four definiti
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