On properties of the solutions to the $\alpha$-harmonic equation

TL;DR

This paper investigates properties of solutions to the $ ext{alpha}$-harmonic equation, establishing inequalities, conditions for composition, and exploring related function spaces, advancing understanding of these generalized harmonic functions.

Contribution

It introduces new inequalities, characterizes composition properties, and studies Bergman-type spaces for $ ext{alpha}$-harmonic functions, extending classical harmonic analysis.

Findings

Schwarz and Schwarz-Pick type inequalities derived

Conditions for composition with analytic functions established

Bergman-type spaces for $ ext{alpha}$-harmonic functions analyzed

Abstract

The aim of this paper is to establish properties of the solutions to the $\alpha$-harmonic equations: $\Delta_{\alpha}(f(z))=\partial{z}[(1-{|{z}|}^{2})^{-\alpha} \overline{\partial}{z}f](z)=g(z)$, where $g:\overline{\mathbb{ID}}\rightarrow\mathbb{C}$ is a continuous function and $\overline{\mathbb{D}}$ denotes the closure of the unit disc $\mathbb{D}$ in the complex plane $\mathbb{C}$. We obtain Schwarz type and Schwarz-Pick type inequalities for the solutions to the $\alpha$-harmonic equation. In particular, for $g\equiv 0$, the solutions to the above equation are called $\alpha$-harmonic functions. We determine the necessary and sufficient conditions for an analytic function $\psi$ to have the property that $f\circ\psi$ is $\alpha$-harmonic function for any $\alpha$-harmonic function $f$. Furthermore, we discuss the Bergman-type spaces on $\alpha$-harmonic functions.