Analysis 33, 143–157 (2013) / DOI 10.1524/anly.2013.1185 c  Oldenbourg Wissenschaftsverlag, M¨ unchen 2013 Meromorphic functions whose certain differential polynomials share a small function with finite weight Pulak Sahoo  , Sajahan Seikh Received: December 3, 2012 Summary: With the notion of weighted sharing of values we study the uniqueness of meromorphic functions whose certain nonlinear differential polynomials share a small function. Our result improves a recent result due to A. Banerjee [Kyungpook Math. J., 51 (2011), 43–58] and that of the results of J. Xia and Y. Xu [Filomat J. Math., 25 (2011), 185–194]. 1 Introduction, definitions and results In this paper, by meromorphic functions we will always mean meromorphic functions in the complex plane. We adopt the standard notations in the Nevanlinna theory of meromorphic functions as explained in [4], [15] and [16]. 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 nonconstant 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/ D o¹T .r; h/º.r !1;r 62 E/: 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/ D o¹T.r/º .r !1;r 62 E/: Let f and g be two nonconstant meromorphic functions. A meromorphic function ˛.z/ is said to be a small function of f , provided T .r; ˛/ D S.r;f /. For a 2 C [ ¹1º we say that f and g share the value a CM (counting multiplicities) if f  a and g  a have the same set of zeros with the same multiplicities and we say that f and g share the value a IM (ignoring multiplicities) if we do not consider the multiplicities. Throughout this paper, we need the following definitions. ‚.a; f / D 1  lim sup r !1 N .r; aI f/ T .r; f / ;  Corresponding author: Pulak Sahoo The first author is thankful to DST-PURSE programme for financial assistance. AMS 2010 subject classifications. Primary 30D35 Key words and phrases: Meromorphic function, small function, weighted sharing, differential polynomials Brought to you by | Penn State - The Pennsylvania State University Authenticated | 128.118.88.48 Download Date | 10/4/13 10:30 PM