Properties for ($α,β$)-harmonic functions

Abstract

We investigate properties of ($α,β$)-harmonic functions. First, we discuss the coefficient estimates for ($α,β$)-harmonic functions. In particular, we obtain Heinz's inequality for ($α,β$)-harmonic functions, propose a coefficient bound for normalized univalent ($α,β$)-harmonic functions and prove that this holds for the subclass that consists of starlike functions. Furthermore, by utilizing the relationship between ($α,β$)-harmonic functions and harmonic functions, we obtain Radó's theorem, Koebe type covering theorems and an area theorem. Finally, we show growth estimates and distortion estimates for ($α,β$)-harmonic functions by using the $L^p$ norms of the boundary functions.