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Schwarz lemma for real harmonic functions onto surfaces with non-negative Gaussian curvature
Published online by Cambridge University Press: 15 June 2023
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
MSC classification
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- Research Article
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- © The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.