Invariant gradient in refinements of Schwarz and Harnack inequalities

Abstract

In this paper we prove a refinement of Schwarz's lemma for holomorphic mappings from the unit ball Bn ⊂ Cn to the unit disk D ⊂ C obtained by Kalaj in [3]. We also give some corollaries of this result and a similar result for pluriharmonic functions. In particular, we give an improvement of Schwarz's lemma for non-vanishing holomorphic functions from Bn to D that was obtained in a recent paper by Dyakonov [2]. Finally, we give a new and short proof of Markovic's theorem on contractivity of harmonic mappings from the upper half-plane H to the positive reals. The same result does not hold for higher dimensions, as is shown by given counterexamples.