An invitation to the theory of geometric functions

TL;DR

This paper introduces the fundamentals of geometric function theory, focusing on univalent and Caratheodory functions, aiming to guide young researchers through basic concepts, techniques, and examples in the field.

Contribution

It provides an accessible overview of key concepts and techniques in geometric functions, especially relating Caratheodory functions to other classes, tailored for beginners and aspiring researchers.

Findings

Explains basic terminologies and concepts in geometric functions.

Highlights techniques for establishing results using Caratheodory functions.

Includes examples demonstrating the application of these techniques.

Abstract

This note is an invitation to the theory of geometric functions. The foundation techniques and some of the developments in the field are explained with the mindset that the audience is principally young researchers wishing to understand some basics. It begins with the basic terminologies and concepts, then a mention of some subjects of inquiry in univalent functions theory. Some of the most basic subfamilies of the family of univalent functions are mentioned. Main emphasy is on the important class of Caratheodory functions and their relations with the various classes of functions, especially the techniques for establishing results in those other classes when compared with the underlying Caratheodory functions. This is contained in Section 4. Examples based on this technique are given in the last section. Since the target audience is the uninitiated, the difficult proofs are not presented. The elementary proofs are explained in the simplest terms. Footnotes are made to further explain some not-immediately obvious points. The references are mostly standard texts. The interested may consult experts for the most recent references in addition to those contained in the cited texts. Hopefully, this may as well profit even the initiated who intends to research in this field.