We employ the idea of weighted sharing of sets to find a unique range set for meromorphic functions with deficient values. Our result improves, generalises, and extends the result of Lahiri. Examples are exhibited that a condition in one of our results is the best possible one. 1. Introduction Definitions and Results In this paper by meromorphic functions we will always mean meromorphic functions in the complex plane. It will be convenient to let denote any set of positive real numbers of finite linear measure, not necessarily the same at each occurrence. For any nonconstant meromorphic function we denote by any quantity satisfying We denote by the maximum of and . The notation denotes any quantity satisfying as , . We adopt the standard notations of the Nevanlinna theory of meromorphic functions as explained in [1]. For , we define Let and be two nonconstant meromorphic functions, and let be a finite complex number. We say that and share CM, provided that and have the same zeros with the same multiplicities. Similarly, we say that and share IM, provided that and have the same zeros ignoring multiplicities. In addition we say that and share CM, if and share CM, and we say that and share IM, if and share IM. Let be a set of distinct elements of and , where each point is counted according to its multiplicity. Denote by the reduced form of . If , we say that and share the set CM. On the other hand if , we say that and share the set IM. Evidently, if contains only one element, then it coincides with the usual definition of CM (resp., IM) shared values. Let a set and and be two nonconstant meromorphic (entire) functions. If implies , then is called a unique range set for meromorphic (entire) functions or in brief URSM (URSE). We will call any set a unique range set for meromorphic functions ignoring multiplicity (URSM-IM) for which implies for any pair of nonconstant meromorphic functions. Inspired by Nevanlinna’s 5 and 4 value theorem, in [2, 3], Gross raised the problem of finding out a finite set so that an entire function in the complex plane is determined by the preimage of , where each pre-image of related to some entire function is counted according to its multiplicity. In 1982 Gross and Yang [4] proved the following theorem. Theorem A. Let . If two entire functions and satisfy , then . Noting that the set in Theorem A is an infinite set, we know that Theorem A does not give a solution to the problem of Gross. In 1994 Yi [5] exhibited a URSE with 15 elements, and in 1995 Li and Yang [6] exhibited a URSM with 15 elements and a URSE with 7 elements.
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