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Normal Functions: Lp Estimates

Published online by Cambridge University Press:  20 November 2018

Huaihui Chen  and
Paul M. Gauthier
Huaihui Chen
Affiliation:
Department of Mathematics, Nanjing Normal University, Nanjing, Jiangsu 210024, P.R. China
Paul M. Gauthier
Affiliation:
Département de Mathématiques et de Statistique, Université de Montréal, Montréal, Québec H3C 3J7, Canada
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Abstract

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For ameromorphic (or harmonic) function ƒ, let us call the dilation of ƒ at z the ratio of the (spherical)metric at ƒ(z) and the (hyperbolic)metric at z. Inequalities are knownwhich estimate the sup norm of the dilation in terms of its Lp norm, for p > 2, while capitalizing on the symmetries of ƒ. In the present paper we weaken the hypothesis by showing that such estimates persist even if the Lp norms are taken only over the set of z on which ƒ takes values in a fixed spherical disk. Naturally, the bigger the disk, the better the estimate. Also, We give estimates for holomorphic functions without zeros and for harmonic functions in the case that p = 2.

Keywords

30D4530F35
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 1 , 01 February 1997 , pp. 55 - 73
Copyright
Copyright © Canadian Mathematical Society 1997

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