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

2020 ◽  
Vol 14 (1) ◽  
pp. 150-168
Author(s):  
Miodrag Mateljevic ◽  
Marek Svetlik
Keyword(s):  
Simple Proof ◽  
Hyperbolic Geometry ◽  
Harmonic Functions ◽  
Hyperbolic Metric ◽  
Schwarz Lemma ◽  
Optimal Estimates ◽  
The Schwarz Lemma ◽  
The Way

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.

scholarly journals On harmonic functions and the Schwarz lemma

2011 ◽  
Vol 140 (1) ◽  
pp. 161-165 ◽  
Author(s):  
David Kalaj ◽  
Matti Vuorinen
Keyword(s):  
Harmonic Functions ◽  
Schwarz Lemma ◽  
The Schwarz Lemma

scholarly journals A note on the Schwarz lemma for harmonic functions

Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3711-3720
Author(s):  
Marek Svetlik
Keyword(s):  
Unit Disk ◽  
Harmonic Functions ◽  
Schwarz Lemma ◽  
The Schwarz Lemma

In this note we consider some generalizations of the Schwarz lemma for harmonic functions on the unit disk, whereby values of such functions and the norms of their differentials at the point z = 0 are given.

scholarly journals On the Schwarz lemma at the upper half plane

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 2995-3011
Author(s):  
Bülent Örnek
Keyword(s):  
Holomorphic Function ◽  
Real Axis ◽  
Half Plane ◽  
Simple Proof ◽  
Boundary Point ◽  
Schwarz Lemma ◽  
The Real ◽  
Upper Half Plane ◽  
Boundary Schwarz Lemma ◽  
The Schwarz Lemma

In this paper, we give a simple proof for the boundary Schwarz lemma at the upper half plane. Considering that f(z) is a holomorphic function defined on the upper half plane, we derive inequalities for the modulus of derivative of f (z), |f'(0)| by assuming that the f(z) function is also holomorphic at the boundary point z = 0 on the real axis with f(0)=Rf(i).

scholarly journals SCHWARZ LEMMA FOR HOLOMORPHIC MAPPINGS IN THE UNIT BALL

2017 ◽  
Vol 60 (1) ◽  
pp. 219-224 ◽  
Author(s):  
DAVID KALAJ
Keyword(s):  
Unit Ball ◽  
Harmonic Functions ◽  
Type Inequality ◽  
Holomorphic Mappings ◽  
Schwarz Lemma ◽  
Complex Spaces

AbstractIn this note, we establish a Schwarz–Pick type inequality for holomorphic mappings between unit balls Bn and Bm in corresponding complex spaces. We also prove a Schwarz-Pick type inequality for pluri-harmonic functions.

Extension of the Schwarz Lemma to homeomorphisms with controlled p-module

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
Anatoly Golberg ◽  
Ruslan Salimov
Keyword(s):  
Schwarz Lemma ◽  
The Schwarz Lemma
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