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We study the analytic characteristic properties for the sense preserving harmonic mappings and quasiconformal mappings on the plane.Including some extremal problems for harmonic mappings such as Landau Theorem and Blochs Theorem in harmonic mappings,univalent harmonic mappings in the given domain to be harmonic K-quasiconformal mappings,the stable properties for univalent harmonic mappings and some relative problems.