Abstract
We study the analytic characteristic property for the sense-preserving univalent harmonic mappings on the upper half-plane onto itself. Using the representing formula for positive harmonic mappings on the upper half-plane, one necessary and sufficient condition for sense-preserving univalent harmonic mappings on the upper half-plane onto itself to be harmonic quasiconformal mappings is obtained. As an application, by the property of the two-sided univalent harmonic mappings on the upper half-plane onto itself, we prove that the set of univalent harmonic quasiconformal mappings on the upper half-plane onto itself with respect to composition is not a group.
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