Schwarz-Pick lemma for (α,β)-harmonic functions in the unit disc

Abstract

We obtain Schwarz–Pick lemma for (α,β)-harmonic functions u in the disc, where α and β are complex parameters satisfying ℜα+ℜβ>−1. We prove sharp estimate of ‖Du(0)‖ for such functions in terms of Lp norm of the boundary function. Also, we give asymptotically sharp estimate of ‖Du(z)‖. This estimate is generalized to higher order derivatives. Our results extend earlier results for α-harmonic and Tα-harmonic functions.