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Schwarz lemma for real harmonic functions onto surfaces with non-negative Gaussian curvature.
- Source :
- Proceedings of the Edinburgh Mathematical Society; May2023, Vol. 66 Issue 2, p516-531, 16p
- Publication Year :
- 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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00130915
- Volume :
- 66
- Issue :
- 2
- Database :
- Complementary Index
- Journal :
- Proceedings of the Edinburgh Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 165473791
- Full Text :
- https://doi.org/10.1017/S0013091523000263