Quantum Approach in Geometric Function Theory

Geometric function theory and special functions have developed into two rich and highly interconnected areas of modern mathematical analysis. Classical results concerning analytic and univalent functions have played a central role in complex analysis, while special functions such as the Gamma, Beta, and hypergeometric functions have provided powerful analytic tools with applications ranging from number theory to mathematical physics. In recent decades, the emergence of $q$-analysis, also known as quantum calculus, has led to a natural extension of many classical concepts by introducing a deformation parameter $q$. This book replaces classical derivatives and integrals with their $q$-analogues and allows the development of new families of special functions and analytic structures. The primary aim of this book is to present a systematic study of geometric function theory in the setting of quantum calculus. In particular, the authors focus on the role of $q$-special functions and their applications to univalent and harmonic mappings.

This book begins with the classical foundations and gradually develops the theory toward modern topics in quantum calculus. Our goal is to provide the necessary analytic background together with the tools required to explore recent developments in the quantum extension of geometric function theory. This book is organized into ten chapters that develop the theory in a progressive manner.

This book is primarily intended for graduate students, researchers, and academics in the fields of complex analysis, geometric function theory, special functions, and quantum calculus. It may also be used as a supporting text in advanced courses and as a resource for research projects.