An over view of geometric function theory (GFT) and analytic properties of meromorphic functions

This article presents an over view of Geometric Function Theory (GFT) and utilized the analytic properties of meromorphic functions. Geometric Function Theory is a branch of mathematics that focuses on the geometric interpretations and implications of analytic functions defined in the complex plane. Our exploration begins with an itemized discussion of key concepts within GFT, emphasizing their relevance and theoretical underpinnings. Central to our study is the investigation of meromorphic functions, which are functions that are analytic except for isolated singularities where they may have poles. We examined various classes of meromorphic functions and elucidate their properties, including their behavior near singularities and their broader geometric implications. A significant portion of our inquiry involves the Hadamard product of functions. This operation allows us to explore the combined effect of two analytic functions, considering their series expansions and how their product transforms under this operation. By studying the Hadamard transformation, we uncover analogues and interesting results that shed light on the interplay between analytic functions and their geometric representations. We also provide detailed diagrammatic descriptions of fundamental geometric shapes such as circles, open unit disks, and closed unit disks. These diagrams serve to visually illustrate key concepts and relationships within GFT, aiding in the understanding of how analytic functions behave in different spatial configurations. Our article offers a comprehensive exploration of Geometric Function Theory, emphasizing its foundational concepts and their applications in analyzing analytic and meromorphic functions.