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Conformal invariants and higher-order schwarz lemmas

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Abstract

We derive a generalization of the Grunsky inequalities using the Dirichlet principle. As a corollary, sharp distortion theorems for bounded univalent functions are proven for invariant differential expressions which are higher-order versions of the Schwarzian derivative. These distortion theorems can be written entirely in terms of conformai invariants depending on the derivatives of the hyperbolic metric, and can be interpreted as ’Schwarz lemmas’. In particular, sharp estimates on distortion of the derivatives of geodesic curvature of a curve under bounded univalent maps are given.

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  1. Department of Mathematics, University of Michigan East Hall, 525 East University Ave, 48109-1109, Ann Arbor, MI, USA

    Eric Schippers

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  1. Eric Schippers
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Schippers, E. Conformal invariants and higher-order schwarz lemmas. J. Anal. Math. 90, 217–241 (2003). https://doi.org/10.1007/BF02786557

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  • DOI: https://doi.org/10.1007/BF02786557

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