SOME RESULTS ON THE QUESTIONS OF KIT-WING YU

Abstract

Abstract. The paper deals with the problem of meromorphic functionssharing a small function with its differential polynomials and improvesthe results of Liu and Gu [9], Lahiri and Sarkar [8], Zhang [13] and Zhangand Yang [14] and also answer some open questions posed by Kit-Wing Yu[16]. In this paper we provide some examples to show that the conditionsin our results are the best possible. 1. Introduction, definition and resultsIn this paper by meromorphic functions we will always mean meromorphicfunctions in the complex plane.Let f and g be two non-constant meromorphic functions and let a be a finitecomplex number. We say that f and g share a CM, provided that f − a andg − a have the same zeros with the same multiplicities. Similarly, we say thatf and g share a IM, provided that f −a and g−a have the same zeros ignoringmultiplicities. In addition we say that f and g share ∞ CM, if 1/f and 1/gshare 0 CM, and we say that f and g share ∞ IM, if 1/f and 1/g share 0 IM.We adopt the standard notations in Nevanlinna’s value distribution theoryof meromorphic functions such as the characteristic function T(r,f), the count-ing function of the poles N(r,∞;f) and the proximity function m(r,∞;f) (see[10]). For a non-constant meromorphic function f we denote by S(r,f) anyquantity satisfying S(r,f) = o(T(r,f)) as r → ∞, outside of a possible ex-ceptional set of finite linear measure. Let k ∈ N and a ∈ C∪ {∞}. We useN