Absolutely Convex, Uniformly Starlike and Uniformly Convex Harmonic Mappings

TL;DR

This paper characterizes absolutely convex harmonic functions, establishes coefficient bounds, growth, covering, and area theorems, and explores their connections with starlike and convex functions, including applications to hypergeometric and polylogarithm functions.

Contribution

It provides necessary and sufficient conditions for absolute convexity in harmonic functions and explores their relationships with other classes, extending classical operator theory.

Findings

Established coefficient bounds for absolutely convex harmonic mappings.

Proved growth, covering, and area theorems for these mappings.

Connected harmonic function classes with hypergeometric and polylogarithm functions.

Abstract

In this paper, we prove necessary and sufficient conditions for a sense-preserving harmonic function to be absolutely convex in the open unit disk. We also estimate the coefficient bound and obtain growth, covering and area theorems for absolutely convex harmonic mappings. A natural generalization of the classical Bernardi type operator for harmonic functions is considered and its connection between certain classes of uniformly starlike harmonic functions and uniformly convex harmonic functions is also investigated. At the end, as applications, we present a number of results connected with hypergeometric and polylogarithm functions.