The Boundary Schwarz lemma for harmonic and pluriharmonic mappings and some generalisations

TL;DR

This paper extends Schwarz lemmas to harmonic and pluriharmonic mappings in Hilbert and Banach spaces, providing new boundary results and generalizations that enhance understanding of these mappings in high-dimensional spaces.

Contribution

It develops methods to generalize classical Schwarz lemmas to harmonic and pluriharmonic mappings in infinite-dimensional spaces, with applications to boundary lemmas and mappings between Euclidean balls.

Findings

Extended Schwarz lemmas to Hilbert and Banach spaces.

Improved boundary Schwarz lemmas for pluriharmonic mappings.

Results on harmonic mappings between Euclidean balls.

Abstract

The main purpose of this paper is to develop some methods to investigate the Schwarz type lemmas of holomorphic mappings and pluriharmonic mappings in Hilbert and Banach spaces. Initially, we extend the classical Schwarz lemmas of holomorphic mappings to Hilbert and Banach spaces. Furthermore, we improve and generalize the classical Schwarz lemmas of planar harmonic mappings into the sharp forms of Banach spaces, and present some applications to sharp boundary Schwarz type lemmas for pluriharmonic mappings in Hilbert and Banach spaces and the obtained results provide the improvements and generalizations of some latest corresponding results in this subject. In the last section we studied harmonic and hyperbolical-harmonic function mapping unit ball in Euclidial n-space, to unit ball in Euclidial m-space, taking prescribed value in the origin, which improved some previous results.