DOI: 10.1515/ms-2015-0126 Math. Slovaca 66 (2016), No. 1, 159–168 UNIQUENESS OF MEROMORPHIC FUNCTIONS AND NONLINEAR DIFFERENTIAL POLYNOMIALS SHARING A NONZERO POLYNOMIAL Pulak Sahoo* — Sajahan Seikh** (Communicated by Stanis lawa Kanas ) ABSTRACT. In the paper, we study the uniqueness of meromorphic functions when certain nonlinear differential polynomials share a nonzero polynomial. The results of the paper improves two recent results due to [LI, X. M.—YI, H. X.: Uniqueness of meromorphic functions whose certain nonlinear differential polynomials share a polynomial, Comput. Math. Appl. 62 (2011), 539–550]. c 2016 Mathematical Institute Slovak Academy of Sciences 1. Introduction, definitions and results In this paper, by meromorphic functions we will always mean meromorphic functions in the complex plane. We assume the reader is familiar with the basic notions of Nevanlinna value distribution theory (see [6], [15] and [16]). For a nonconstant meromorphic function f , we denote by T (r, f ) the Nevanlinna characteristic of f and by S(r, f ) any quantity satisfying S(r, f )= o{T (r, f )} as r →∞ outside of an exceptional set of finite linear measure. We denote by T (r) the maximum of T (r, f ) and T (r, g). The symbol S(r) denotes any quantity satisfying S(r)= o{T (r)} as r →∞. Let f and g be two nonconstant meromorphic functions and a ∈ C ∪ {∞}. If f - a and g - a have the same zeros, we say that f and g share the value a IM (ignoring multiplicities). If f - a and g - a have the same zeros with the same multiplicities, then we say that f and g share the value a CM (counting multiplicities). In addition, we need the following definitions. 1 ([9]) Let a ∈ C ∪ {∞}. We denote by N (r, a; f |= 1) the counting function of simple a points of f . For a positive integer p we denote by N (r, a; f |≤ p) the counting function of those a-points of f (counted with proper multiplicities) whose multiplicities are not greater than p. By N (r, a; f |≤ p) we denote the corresponding reduced counting function. Analogously we can define N (r, a; f |≥ p) and N (r, a; f |≥ p). 2 ([8]) Let k be a positive integer or infinity. We denote by N k (r, a; f ) the counting function of a-points of f , where an a-point of multiplicity m is counted m times if m ≤ k and k times if m>k. Then N k (r, a; f )= N (r, a; f )+ N (r, a; f |≥ 2) + ··· + N (r, a; f |≥ k). Clearly N 1 (r, a; f )= N (r, a; f ). 2010 Mathematics Subject Classification: Primary 30D35. K e y w o r d s: uniqueness, meromorphic function, nonlinear differential polynomials. 159 Unauthenticated Download Date | 5/17/16 9:48 AM