On some Schwarz type inequalities

Miodrag Mateljević,Adel Khalfallah

doi:10.1186/s13660-020-02433-6

Abstract

First, we establish some Schwarz type inequalities for mappings with bounded Laplacian, then we obtain boundary versions of the Schwarz lemma.

Highlights

Introduction and preliminariesMotivated by the role of the Schwarz lemma in complex analysis and numerous fundamental results, see for instance [16, 19] and references therein, in 2016, the first author [1](a) has posted on ResearchGate the project “Schwarz lemma, the Carathéodory and Kobayashi Metrics and Applications in Complex Analysis”.a 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?” [1](b).b In this project and in [16], cf. [13], we developed the method related to holomorphic mappings with strip codomain
Theorem 1 Let f : U → U be a harmonic function from the unit disc into itself
We provide a different and an elementary proof of Theorem C, giving a Schwarz type lemma for mappings satisfying Poisson’s equations

Summary

Introduction and preliminaries

Motivated by the role of the Schwarz lemma in complex analysis and numerous fundamental results, see for instance [16, 19] and references therein, in 2016, the first author [1](a) has posted on ResearchGate the project “Schwarz lemma, the Carathéodory and Kobayashi Metrics and Applications in Complex Analysis”.a 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?” [1](b).b In this project and in [16], cf. [13], we developed the method related to holomorphic mappings with strip codomain (we refer to this method as the approach via the Schwarz–Pick lemma for holomorphic maps from the unit disc into a strip). Mateljević and Sveltik [18] proved a Schwarz lemma for real harmonic functions with values in (–1, 1) using a completely different approach than Burgeth [3] and showed that the inequalities obtained are sharp. Mateljević and Sveltik [18] proved a Schwarz lemma for real harmonic functions with values in (–1, 1) using a completely different approach from Burgeth [3]. We can extend Theorem B for complex harmonic functions from the unit disc into itself. Theorem 1 Let f : U → U be a harmonic function from the unit disc into itself.

Then f and f

The estimate