Uniqueness of $L$ function with special class of meromorphic function under restricted sharing of sets

TL;DR

This paper introduces a stronger form of set sharing called restricted set sharing to address errors in previous work and applies it to solve a longstanding problem in the uniqueness of the $L$ function for certain meromorphic functions.

Contribution

It proposes a new concept of restricted set sharing and demonstrates its effectiveness in resolving errors and a major open problem in the uniqueness theory of meromorphic functions.

Findings

Resolved previous errors in the literature.

Introduced and validated the concept of restricted set sharing.

Solved a long-standing open problem in the field.

Abstract

The purpose of the paper is to rectify a series of errors occurred in [2], [17], [20] for a particular situation. To get a fruitful solution and to overcome the issue, we introduce a new form of set sharing namely restricted set sharing, which is stronger than the usual one. We manipulate the newly introduced notion in this specific section of literature to resolve all the complications. Not only that we have subtly used the same sharing form to a well known unique range set [3] to settle a long time unsolved problem.