Properties for ($α,β$)-harmonic functions
Jinjing Qiao, Jiale Chang, Antti Rasila
TL;DR
This work extends the theory of harmonic mappings by studying ($\alpha,\beta$)-harmonic functions, defined via the elliptic operator $L_{\alpha,\beta}$ and the associated Poisson kernel. It derives Heinz-type coefficient inequalities and establishes Radó-type univalence, Koebe-type covering, and area theorems for the ($\alpha,\beta$)-harmonic family, with explicit gamma- and hypergeometric-based constants that recover classical results in the limit $\alpha,\beta\to0$. The authors also develop growth and distortion estimates in terms of $L^{p}$ norms of boundary data, providing sharp (asymptotically) bounds for the function and its first-order derivatives, and unifying several known results for harmonic and $\alpha$-harmonic mappings. The results have implications for geometric function theory of generalized harmonic mappings and offer tools for quantitative distortion and area bounds in this broader setting.
Abstract
We investigate properties of ($α,β$)-harmonic functions. First, we discuss the 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 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.