Some Aspects of Uniqueness Theory of Entire and Meromorphic Functions (Ph.D. thesis)

TL;DR

This thesis investigates the uniqueness of meromorphic functions using Nevanlinna theory, focusing on Bruck's conjecture, set sharing, and new unique range sets, advancing the understanding of function uniqueness and related conjectures.

Contribution

It introduces new types of unique range sets, explores their properties, and connects Bruck's conjecture with Gross' Problem, improving existing results in the field.

Findings

Established new conditions for function uniqueness

Connected Bruck's conjecture with Gross' Problem

Provided examples demonstrating sharpness of conditions

Abstract

The subject of our thesis is the uniqueness theory of meromorphic functions and it is devoted to problems concerning Bruck conjecture, set sharing and related topics. The tool, we used in our discussions is classical Nevanlinna theory of meromorphic functions. In 1996, in order to find the relation between an entire function with its derivative, counterpart sharing one value CM, a famous conjecture was proposed by R. Bruck. Since then the conjecture and its analogous results have been investigated by many researchers and continuous efforts have been put on by them. In our thesis, we have obtained similar types of conclusions as that of Bruck for two differential polynomials which in turn improve several existing results under different sharing environment. A number of examples have been exhibited to justify the necessity or sharpness of some conditions, hypothesis used in the thesis. As a variation of value sharing, F. Gross first introduced the idea of set sharing, by proposing a problem, which has later became popular as Gross Problem. Inspired by the Gross' Problem, the set sharing problems were started which was later shifted towards the characterization of the polynomial backbone of different unique range sets. In our study, we introduced some new type of unique range sets and at the same time, we further explored the anatomy of these unique range sets generating polynomials as well as connected Bruck conjecture with Gross' Problem.