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

The aim of this paper is to establish properties of the solutions to the $\alpha$-harmonic equations: $\Delta_{\alpha}(f(z))=\partial{z}(1-{|{z}|}^{2})^{-\alpha} \overline{\partial}{z}f=g(z)$,where$g:\overline{$\mathbb{I}\mathbb{D}$}\rightarrow$\mathbb{C}$$isacontinuousfunctionand$\overline{$\mathbb{D}$}$denotestheclosureoftheunitdisc$$\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.