An Introduction to Univalent Functions

This book introduces the theory of univalent functions, focusing on the key subclasses of starlike, convex and close-to-convex functions. These subclasses play a central role in the study of geometric function theory. The book begins with the general class of univalent functions and studies their basic properties. It then explores essential mathematical tools, including results on growth, distortion and coefficients for bounded analytic functions, as well as the theory of functions with positive real parts and differential subordination. The last three chapters delve into starlike, convex and close-to-convex functions, presenting key results on coefficient bounds, growth, distortion estimates and subordination theorems.
Designed for graduate students and early-stage researchers, the book requires minimal prerequisites of a basic course in complex analysis. Detailed proofs have been provided for all results, ensuring that readers can follow the logical development of the material. Exercises given at the end of each chapter offer opportunities for further study and deeper understanding of the concepts. This book serves as both a textbook and a resource for those beginning research in geometric function theory. Its accessible style, coupled with its clear explanations and motivating examples, makes it an ideal choice for anyone looking to explore the fascinating world of univalent functions.