On harmonic functions on surfaces with positive Gauss curvature and the Schwarz lemma

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.

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A note on the Schwarz lemma for harmonic functions

Filomat ◽

10.2298/fil2011711s ◽

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.

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Generalized harmonic functions and Schwarz lemma for biharmonic mappings

Monatshefte für Mathematik ◽

10.1007/s00605-021-01619-4 ◽

2021 ◽

Author(s):

Adel Khalfallah ◽

Fathi Haggui ◽

Mohamed Mhamdi

Keyword(s):

Harmonic Functions ◽

Schwarz Lemma ◽

Generalized Harmonic Functions

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SCHWARZ LEMMA FOR HOLOMORPHIC MAPPINGS IN THE UNIT BALL

Glasgow Mathematical Journal ◽

10.1017/s0017089517000052 ◽

2017 ◽

Vol 60 (1) ◽

pp. 219-224 ◽

Cited By ~ 1

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.

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