Abstract
The aim of this paper is to establish properties of solutions to the α-harmonic equations: , where is a continuous function and denotes the closure of the unit disc in the complex plane . We obtain Schwarz type and Schwarz-Pick type inequalities for solutions to the α-harmonic equation. In particular, for , solutions to the above equation are called α-harmonic functions. We determine the necessary and sufficient conditions for an analytic function ψ to have the property that is α-harmonic function for any α-harmonic function f. Furthermore, we discuss the Bergman-type spaces on α-harmonic functions.
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