Rational function and differential polynomial of a meromorphic function sharing a small function

Abstract

In this paper, we have mainly dealt with the relation between a generalized differential polynomial and a rational function R(f) of a non-constant meromorphic function f sharing a small function a ≡ a(z) (≢ 0,∞). Our results will extend recent results in [4], [5], and [9] in the direction of Bruck Conjecture. We have exhibited some examples which show that the result of this paper may or may not be true because non-constant entire functions and conditions obtained in the theorems cannot be removed. Other examples have also substantiated our certain claims.