Hyperbolic metric on the strip and the Schwarz lemma for HQR mappings

Miodrag Mateljevic,Marek Svetlik

doi:10.2298/aadm200104001m

Miodrag Mateljevic, Marek Svetlik

Open Access

https://doi.org/10.2298/aadm200104001m

Copy DOI

Abstract

We give simple proofs of various versions of the Schwarz lemma for real valued harmonic functions and for holomorphic (more generally harmonic quasiregular, shortly HQR) mappings with the strip codomain. Along the way, we get a simple proof of a new version of the Schwarz lemma for real valued harmonic functions (without the assumption that 0 is mapped to 0 by the corresponding map). Using the Schwarz-Pick lemma related to distortion for harmonic functions and the elementary properties of the hyperbolic geometry of the strip we get optimal estimates for modulus of HQR mappings.

Highlights

Motivated by the role of the Schwarz lemma in complex analysis and numerous fundamental results, in 2016, cf. [33](a), the first author has posted the current research project “Schwarz lemma, the Caratheodory and Kobayashi Metrics and Applications in Complex Analysis”∗
Various discussions regarding the subject can be found in the Q&A section on ResearchGate under the question “What are the most recent versions of the Schwarz lemma ?”,[33](b)†. In this project and in [25], cf. [13] we developed the method related to holomorphic mappings with strip codomain
Note here that our use of terms the Schwarz lemma and the Schwarz-Pick lemma is refer to the corresponding versions for modulus and hyperbolic distances, respectively

Summary

INTRODUCTION

Motivated by the role of the Schwarz lemma in complex analysis and numerous fundamental results (see [1, 32, 14, 25] and references cited there and for some recent result which are in our research direction [2, 13, 15, 20, 36]), in 2016, cf. [33](a), the first author has posted the current research project “Schwarz lemma, the Caratheodory and Kobayashi Metrics and Applications in Complex Analysis”∗. Theorem 5 follows directly from the formula (15) of Lemma 1 and the subordination principle Further development of this method yields Theorem 6 (without hypothesis that 0 is mapped to 0) which seems to be a new result (see Example 3 and Lemma 3). Note the above simple method described by the properties (I)-(IV) is basically based on the Schwarz-Pick lemma for holomorphic maps from U into S and it yields a proof of the above proposition to which we refer as the Schwarz-Pick lemma related to distortion for harmonic functions from U into (−1, 1). It is convenient that the properties (I)-(IV) together with Proposition 2 we shortly call the strip property of harmonic functions and refer to it as the strip method By this proposition we control distortion of HQR mappings (see the proof of Lemma 4 below). Note that proof of Lemma 4 is based on Proposition 2 (the Schwarz-Pick lemma related to distortion for harmonic functions from U into (−1, 1))

SOME EXAMPLES AND SOME PROPERTIES OF THE STRIP

EUCLIDEAN PROPERTIES OF HYPERBOLIC DISCS

THE SCHWARZ LEMMA FOR HOLOMORPHIC MAPS FROM U INTO S

THE SCHWARZ LEMMA FOR HARMONIC K-QUASIREGULAR MAPS FROM U INTO S