The Schwarz Lemma and Hyperbolic Geometry

2001 ◽  
pp. 260-273
Author(s):  
Theodore W. Gamelin
Keyword(s):  
Hyperbolic Geometry ◽  
Schwarz Lemma ◽  
The Schwarz Lemma

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.

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

scholarly journals Applications of the Jack's lemma for the meromorphic functions at the boundary

2019 ◽  
Vol 38 (7) ◽  
pp. 219-226
Author(s):  
Tugba Akyel ◽  
Bulent Nafi Ornek
Keyword(s):  
Boundary Point ◽  
Meromorphic Functions ◽  
Schwarz Lemma ◽  
Jack’S Lemma ◽  
Boundary Version ◽  
Jack's Lemma ◽  
The Schwarz Lemma

In this paper, a boundary version of the Schwarz lemma for the class $\mathcal{% N(\alpha )}$ is investigated. For the function $f(z)=\frac{1}{z}% +a_{0}+a_{1}z+a_{2}z^{2}+...$ defined in the punctured disc $E$ such that $% f(z)\in \mathcal{N(\alpha )}$, we estimate a modulus of the angular derivative of the function $\frac{zf^{\prime }(z)}{f(z)}$ at the boundary point $c$ with $\frac{cf^{\prime }(c)}{f(c)}=\frac{1-2\beta }{\beta }$. Moreover, Schwarz lemma for class $\mathcal{N(\alpha )}$ is given.

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