ON UNIQUENESS OF MEROMORPHIC FUNCTIONS PARTIALLY SHARING VALUES WITH THEIR SHIFTS

Abstract

In 1926, R. Nevanlinna showed that two distinct nonconstant meromorphic functions  and  on the complex plane  share five distinct values then  on whole ­_­­ If a meromorpic function with hyper-order less than 1 and its shifts  share four distinct values or share partially four small periodic functions in the complex plane, then whether  or not. Our aim is to study uniqueness of such meromorphic functions. For our purpose, we use techniques in Nevanlinna theory by estimating the counting functions and use the property of defect relation of values on the complex plane. Let  be four small periodic functions with period c in the complex plane for . Then we prove a result as folows: Assume that meromorphic function  of hyper-order less than 1 with its shift  share  CM, shares partially  IM and reduced defect of  at is maximal. Then under an appropriate deficiency assumption, for all  Our result is a continuation of previous works of the authors and provides an understanding of the meromorphic functions of hyper-order less than 1.