  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.
