Schwarz-Pick Lemma for $(\alpha ,\beta )$-Harmonic Functions in the Unit Disc

We obtain Schwarz-Pick lemma for $(\alpha ,\beta )$-harmonic functions u in the disc, where $\alpha$ and $\beta$ are complex parameters satisfying $\mathrm{\Re }\alpha +\mathrm{\Re }\beta >-1$. We prove sharp estimate of derivative at the origin for such functions in terms of $L^p$ norm of the boundary function. Also, we give asymptotically sharp estimate of the norm of the derivative at point $z$ in the unit disc. These estimates are extended to higher order derivatives. Our results extend earlier results for $\alpha$-harmonic and $T_{\alpha }$-harmonic functions.