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2010•Journal of Huaqiao UniversityRequires access

Distortion Estimations of the Hyperbolic Jacobians of Harmonic Quasiconformal Mappings

Xingdi Chen

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Abstract

The distortion estimation with respect to the hyperbolic metrics of two classes of harmonic quasiconformal mappings is studied.First,the sharp upper and lower bounds of the hyperbolic Jacobians of Euclidean harmonic quasiconformal mappings from the upper half-plane onto itself and their corresponding extremal functions are given.Secondly,the distortion estimation of hyperbolic Jacobian of hyperbolic quasiconformal mappings are obtained.Finally,the distortion estimation of the above two classes of mappings is applied to study their corresponding distortion theorems about hyperbolic areas.The results show that the above two classes of harmonic quasiconformal mappings are non-explodable.

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What this paper is about

The distortion estimation with respect to the hyperbolic metrics of two classes of harmonic quasiconformal mappings is studied.First,the sharp upper and lower bounds of the hyperbolic Jacobians of Euclidean harmonic quasiconformal mappings from the upper half-plane onto itself and their corresponding extremal functions are given.Secondly,the distortion estimation of hyperbolic Jacobian of hyperbolic quasiconformal mappings are obtained.Finally,the distortion estimation of the above two classes of mappings is applied to study their corresponding distortion theorems about hyperbolic areas.The results show that the above two classes of harmonic quasiconformal mappings are non-explodable.

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Available abstract

The distortion estimation with respect to the hyperbolic metrics of two classes of harmonic quasiconformal mappings is studied.First,the sharp upper and lower bounds of the hyperbolic Jacobians of Euclidean harmonic quasiconformal mappings from the upper half-plane onto itself and their corresponding extremal functions are given.Secondly,the distortion estimation of hyperbolic Jacobian of hyperbolic quasiconformal mappings are obtained.Finally,the distortion estimation of the above two classes of mappings is applied to study their corresponding distortion theorems about hyperbolic areas.The results show that the above two classes of harmonic quasiconformal mappings are non-explodable.

Key concepts: Mathematics, Distortion (music), Jacobian matrix and determinant, Hyperbolic geometry, Mathematical analysis, Quasiconformal mapping, Harmonic, Hyperbolic function

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