The classical Schwarz-Pick lemma and Julia lemma for holomorphic mappings on the unit disk D are generalized to real harmonic mappings of the unit disk, and the results are precise. It is proved that for a harmonic ma...The classical Schwarz-Pick lemma and Julia lemma for holomorphic mappings on the unit disk D are generalized to real harmonic mappings of the unit disk, and the results are precise. It is proved that for a harmonic mapping U of D into the open interval I = （-1, 1）, AU（z）/cosU（z）π/2≤4/π 1/1-｜z｜^2 holds for z E D, where Au（z） is the maximum dilation of U at z. The inequality is sharp for any z E D and any value of U（z）, and the equality occurs for some point in D if and only if U（z） = 4Re {arctan ~a（z）}, z E D, with a M＆bius transformation φa of D onto itself.展开更多
Consider the Teichmuller mapping f associated with in the unit disc D and the class of all quasiconformal mappings in D with the boundary values of f. For a special holomorphic function, the present paper gives the ne...Consider the Teichmuller mapping f associated with in the unit disc D and the class of all quasiconformal mappings in D with the boundary values of f. For a special holomorphic function, the present paper gives the necessary and sufficient condition on, such that f is uniquely extremal among the class. Further, for a general holomorphic function, the anthors suggest the models of the best possible growth conditions on, such that f is extremal or uniquely extremal among the class respectively.展开更多
基金supported by National Natural Science Foundation of China(Grant No.11071083)
文摘The classical Schwarz-Pick lemma and Julia lemma for holomorphic mappings on the unit disk D are generalized to real harmonic mappings of the unit disk, and the results are precise. It is proved that for a harmonic mapping U of D into the open interval I = （-1, 1）, AU（z）/cosU（z）π/2≤4/π 1/1-｜z｜^2 holds for z E D, where Au（z） is the maximum dilation of U at z. The inequality is sharp for any z E D and any value of U（z）, and the equality occurs for some point in D if and only if U（z） = 4Re {arctan ~a（z）}, z E D, with a M＆bius transformation φa of D onto itself.
文摘Consider the Teichmuller mapping f associated with in the unit disc D and the class of all quasiconformal mappings in D with the boundary values of f. For a special holomorphic function, the present paper gives the necessary and sufficient condition on, such that f is uniquely extremal among the class. Further, for a general holomorphic function, the anthors suggest the models of the best possible growth conditions on, such that f is extremal or uniquely extremal among the class respectively.
