A Survey of the Principle and Development of Geometric Function Theory

Abstract      This article serves as an introduction to the theory of geometric functions. Foundational methodologies and certain advancements within the domain are elucidated with the perspective that the primary audience comprises budding scholars eager to grasp fundamental principles. It commences with rudimentary terminologies and principles, followed by an exploration of select topics within the realm of univalent functions theory. Various fundamental subsets within the umbrella of univalent functions are outlined. Particular emphasis is placed on the significant category of Caratheodory functions and their interrelations w th diverse function classes, particularly the methodologies for deriving conclusions in those alternate classes vis-à-vis the underlying Caratheodory functions. Given the intended audience's novice status, intricate proofs are omitted. Instead, elementary demonstrations are articulated using the most straightforward language possible. Footnotes are incorporated to expound upon points that may not be immediately apparent. References primarily consist of canonical texts. Interested parties are encouraged to consult experts for the latest references, supplementing those cited within the mentioned texts. It is hoped that this exposition will prove beneficial to even seasoned researchers venturing into this field. We commence with the fundamental definition and present a few straightforward examples from the realm of univalent functions. Following cursory examination of the existing literature, we overview the advancements achieved in addressing specific challenges within this domain.