Properties for ($\alpha,\beta$)-harmonic functions

TL;DR

This paper explores the properties of ($eta,eta$)-harmonic functions, including coefficient bounds, inequalities, and growth estimates, extending classical harmonic function results to this broader class.

Contribution

It introduces new coefficient estimates, inequalities, and growth/distortion results for ($eta,eta$)-harmonic functions, linking them to classical harmonic function theorems.

Findings

Established Heinz's inequality for ($eta,eta$)-harmonic functions.

Proposed a coefficient bound for normalized univalent ($eta,eta$)-harmonic functions.

Derived growth and distortion estimates using boundary function norms.

Abstract

We investigate properties of ($\alpha,\beta$)-harmonic functions. First, we discuss the coefficient estimates for ($\alpha,\beta$)-harmonic functions. In particular, we obtain Heinz's inequality for ($\alpha,\beta$)-harmonic functions, propose a coefficient bound for normalized univalent ($\alpha,\beta$)-harmonic functions and prove that this holds for the subclass that consists of starlike functions. Furthermore, by utilizing the relationship between ($\alpha,\beta$)-harmonic functions and harmonic functions, we obtain Rad\'{o}'s theorem, Koebe type covering theorems and an area theorem. Finally, we show growth estimates and distortion estimates for ($\alpha,\beta$)-harmonic functions by using the $L^p$ norms of the boundary functions.