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.

Rosihan M. Ali  is a Professor of Mathematics at Universiti Sains Malaysia. He has been with the University for 40 years since the completion of his Ph.D. in Complex Analysis at Texas Tech University. As a research scholar, his works have garnered accolades and national awards from learned societies and organizations. He has served as Dean of the School of Mathematical Sciences and as the University Deputy Vice-Chancellor. He was also an elected member of the University Board of Governors. Within the national scientific community, he is the Editor-in-Chief of the Bulletin of the Malaysian Mathematical Sciences Society, an SCIE-indexed journal as well as a MathSciNet reference list journal. As Malaysia representative, he was an active member on the Council of the Southeast Asian Mathematical Society from 2004--2025. Prof. Ali is also a Fellow of the Academy of Sciences Malaysia and a Fellow of the Malaysian Mathematical Sciences Society.

 See Keong Lee  is an Associate Professor and Deputy Dean (Academic, Career  & International) at the School of Mathematical Sciences, Universiti Sains Malaysia. He obtained his Ph.D.  from Louisiana State University, Baton Rouge, USA, in stochastic analysis. His current research interests lie in geometric function theory.   He has supervised four Ph.D.  and four Master’s students and has published over 40 research papers. He is a member of the American Mathematical Society and a life member of the Malaysian Mathematical Sciences Society (PERSAMA). He also serves as an editor for the Bulletin of the Malaysian Mathematical Sciences Society and the Korean Journal of Mathematics.

 V. Ravichandran  is a Professor and Head of the Department of Mathematics at the National Institute of Technology, Tiruchirappalli. He has held academic positions at Universiti Sains Malaysia as a Post-Doctoral Fellow, Senior Lecturer (2004–2007), and Visiting Professor (2011–2012). He joined the University of Delhi in 2007 and was promoted to Professor in 2014. He served as Head of the Department of Mathematics at the University of Delhi from 2015 to 2018 and as Chair of the Governing Body of Shyama Prasad Mukherji College for Women from 2017 to 2018.

He is the author of textbooks on complex analysis (USM Press), real analysis, partial differential equations, and discrete structures (Springer). He has published over 225 research papers and has supervised 24 Ph.D. and 9 M.Phil. scholars. He currently serves on the editorial boards of the Bulletin of the Malaysian Mathematical Sciences Society, Journal of Analysis, Journal of the Indian Mathematical Society, International Journal of Mathematics and Mathematical Sciences, and Ganita.