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Schwarz lemma for real harmonic functions onto surfaces with non-negative Gaussian curvature

Part of: Riemann surfaces

Published online by Cambridge University Press:  15 June 2023

David Kalaj ,
Miodrag Mateljević  and
Iosif Pinelis
David Kalaj
Affiliation:
Faculty of Natural Sciences and Mathematics, University of Montenegro, Cetinjski put b.b., Podgorica, Montenegro (davidk@ucg.ac.me)
Miodrag Mateljević
Affiliation:
Faculty of mathematics, University of Belgrade, Belgrade, Serbia, Republic of Serbia (miodrag@matf.bg.ac.rs)
Iosif Pinelis
Affiliation:
Department of Mathematical Sciences, Michigan Technological University, Michigan, MI, USA (ipinelis@mtu.edu)
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Abstract

Assume that f is a real ρ-harmonic function of the unit disk $\mathbb{D}$ onto the interval $(-1,1)$, where $\rho(u,v)=R(u)$ is a metric defined in the infinite strip $(-1,1)\times \mathbb{R}$. Then we prove that $|\nabla f(z)|(1-|z|^2)\le \frac{4}{\pi}(1-f(z)^2)$ for all $z\in\mathbb{D}$, provided that ρ has a non-negative Gaussian curvature. This extends several results in the field and answers to a conjecture proposed by the first author in 2014. Such an inequality is not true for negatively curved metrics.

Keywords

harmonic mappingsminimal surfaces

MSC classification

Primary: 30F15: Harmonic functions on Riemann surfaces
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 66 , Issue 2 , May 2023 , pp. 516 - 531
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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