Abstract
We prove that if f, \(|\mathfrak {R}f|<1\), is an analytic mapping on a surface \(\Sigma \) with curvature bounded from below by a constant \(k<0\), and if \(\sigma \) is the hyperbolic distance on the unit disc, we have
where \(d_\Sigma \) is the distance on \(\Sigma \) generated by a conformal metric on \(\Sigma \). On the other hand, if \(|f|<1\), then
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References
Ahlfors, L.V.: An extension of Schwarz’s Lemma. Trans. Amer. Math. Soc. 43, 359–364 (1938)
Ahlfors, L.V.: Conformal Invariants. McGraw-Hill, New York (1973)
Duren, P., Weir, R.: The pseudohyperbolic metric and Bergman spaces in the ball. Trans. Amer. Math. Soc. 359, 63–76 (2007)
Kalaj, D., Vuorinen, M.: On harmonic functions and the Schwarz lemma. Proc. Amer. Math. Soc. 140, 161–165 (2012)
Kobayashi, S.: Hyperbolic Manifolds and Holomorphic Mappings. World Scientific, Singapore (2005)
Marković, M.: Representations for the Bloch type norm of Fréchet differentiable mappings. J. Geom. Anal., to appear
Pavlović, M.: A Schwarz lemma for the modulus of a vector-valued analytic function. Proc. Amer. Math. Soc. 139, 969–973 (2011)
Pavlović, M.: Function Classes on the Unit Disc. An Introduction (2nd ed.). de Gruyter Studies in Mathematics, vol. 52. de Gruyter, Berlin (2019)
Yau, S.T.: A general Schwarz lemma for Kähler manifolds. Amer. J. Math. 100, 197–203 (1978)
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Marković, M. Lipschitz constants for the real part and modulus of analytic mappings on a negatively curved surface. Arch. Math. 116, 61–66 (2021). https://doi.org/10.1007/s00013-020-01516-6
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DOI: https://doi.org/10.1007/s00013-020-01516-6