Investigation on L function uniqueness of entire and meromorphic function using nevanlinna theory

Abstract

Nevanlinna s theory of meromorphic functions occupies the central position of complex analysis, specially of the Value Distribution Theory. In early nineteenth century a famous Finnish mathematician Rolf Nevanlinna developed a systematic theory of meromorphic functions that made a revolutionary change in the approach of Value Distribution Theory. An analytic function that doesn t have any essential singularity in the open complex plane C. Thefocal theme of the value distribution theory is to study the distribution of roots of f(z) a = 0for a meromorphic function f and a complex number a. Its origin goes back to the classical works of nineteenth century like Sokhotskii s theorem, Weierstrass Casorativ theorem, Picard s theorem and many others. At the end of nineteenth and at the beginning of twentieth century further works were carried out by the French school of scholars like Hadamard, Borel Valiron. All these contributions are regarded as classical whereas the method developed by Rolf Nevanlinna, is considered as modern and is dom inating till date. In the theory of differential equations, in many cases, it is impossible to find an explicit solution for a given differential equation. But, Nevanlinna theory of fers an efficient way for this problem. The only requirement is that the solution must be meromorphic sufficiently large near the boundary of the domain. We study some results on uniqueness and value-sharing of entire and meromorphic functions concerning differential polynomials, also some results concerning to homogeneous differential polynomials with weighted sharing and investigate uniqueness problem of entire functions concerning differ ential polynomials share a small functions. Key words: Nevanlinna theory, uniqueness, value sharing, meromorphic function, differ ential polynomial, weighed sharing, shift etc. The thesis is structured into five chapters. Each of the chapters is sectioned to contain i introduction to respective chapter, literature survey, preliminaries with auxiliary lemmas...