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arXiv · 1808.06647

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

Miodrag Mateljević·Marek Svetlik
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Abstract

We give simple proofs of various versions of the Schwarz lemma for real valued harmonic functions and for holomorphic (more generally harmonic quasi\-re\-gu\-lar, shortly HQR) mappings with the strip codomain. Along the way using the principle of subordination and the corresponding conformal mapping, depicted on the Figure 1, we get a simple proof of a new version of the Schwarz lemma for real valued harmonic functions (see Theorems 4 and 5) and Theorem 6 related to holomorphic mappings. Using the Schwarz-Pick lemma related to distortion for harmonic mappings and the elementary properties of the hyperbolic geometry of the strip we prove Lemma 4, which is a key ingredient in the proof of Theorem 7 which yields optimal estimates for modulus of HQR mappings.

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Miodrag Mateljević, Marek Svetlik. 2018-08-20. Hyperbolic metric on the strip and the Schwarz lemma for HQR mappings. https://arxiv.org/abs/1808.06647

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