Some Investigation on Differential Difference Polynomials of Entire and Meromorphic Functions

Abstract

The Nevanlinna theory, a fundamental branch of complex analysis, serves as a profound exploration into the intricate behaviors of meromorphic functions in the complex plane. Originating from investigations into the distribution of values, this theory offers a com prehensive framework to understand the relationship between these functions singulari ties, their zeros, and their broader analytical properties. This abstract provides an overview of Nevanlinna theory s key concepts, such as the Picard theorem and the Nevanlinna characteristic function, which play pivotal roles in understanding the growth and distribution of meromorphic functions. Beyond its theoretical underpinnings, the theory finds extensive applications in diverse mathematical areas, including number theory, differential equations, and the study of Riemann surfaces. Moreover, this abstract highlights the ongoing relevance of Nevanlinna theory in con temporary research by outlining recent developments and open questions. Its continued impact on modern mathematics underscores the theory s significance as a guiding frame work for exploring the complex and rich structures of meromorphic functions and their broader implications across various mathematical disciplines. The thesis is structured into six distinct chapters, each carefully divided into specific sections to provide a comprehensive exploration of the subject matter. In the opening section of every chapter, an introduction sets the stage by offering an overview and context for the upcoming content. Following this, a comprehensive literature survey is presented within each chapter. This entails a meticulous review of existing scholarly works related to the specific focus of the chapter. The subsequent section in each chapter focuses on laying the groundwork with preliminary concepts. Central to each chapter are the main results, presented along with their respective proofs.