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

Abstract

The aim of this paper is to establish properties of the solutions to the $\alpha$-harmonic equations: ${\Delta }_{\alpha }(f(z))=\partial z[(1-{|z|}^2{)}^{-\alpha }\overline{\partial }zf](z)=g(z)$, where $g:\overline{\mathbb{ID}}\to \mathbb{C}$ 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.

16 pages