Classes Associated with Analytic and Harmonic Univalent Functions

Abstract

Analytic and harmonic univalent functions are fundamental concepts in complex analysis and serve as key tools in various areas of mathematics, including complex function theory and geometric function theory. These classes of functions are particularly important in the study of conformal mappings and have widespread applications in physics, engineering, and other fields. In this abstract, we provide a brief overview of these classes and their significance.Analytic univalent functions are holomorphic (analytic) functions that map the unit disk onto a region in the complex plane in a one-to-one and onto manner. They are known for their role in preserving angles and shapes, making them essential in the study of conformal maps. These functions are often characterized by their coefficients in their Taylor series expansion and have important properties related to convexity, starlikeness, and univalency.harmonic univalent functions are solutions to the Laplace equation that are univalent in the unit disk. They arise in potential theory and have intriguing connections to analytic univalent functions through the so-called Grunsky inequalities. Harmonic univalent functions have applications in problems involving electrostatics and fluid dynamics.