Properties for ($\alpha,\beta$)-harmonic functions

Abstract

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