摘要
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.
The classical Schwarz-Pick lemma and Julia lemma for holomorphic mappings on the unit diskD 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),ΛU(z)/cosU(z)π/2≤4/π1/1|z|2 holds for z∈D,whereΛU(z)is the maximum dilation of U at z.The inequality is sharp for any z∈D and any value of U(z),and the equality occurs for some point in D if and only if U(z)=4πRe{arctan(z)},z∈D,with a Mbius transformation of D onto itself.
基金
supported by National Natural Science Foundation of China(Grant No.11071083)
