The Landau theorem and Bloch theorem for planar harmonic and pluriharmonic mappings

Chen, Huaihui; Gauthier, Paul

doi:10.1090/S0002-9939-2010-10659-7

The Landau theorem and Bloch theorem for planar harmonic and pluriharmonic mappings
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by Huaihui Chen and Paul M. Gauthier PDF

Proc. Amer. Math. Soc. 139 (2011), 583-595 Request permission

Abstract:

For a normalized quasiregular pluriharmonic mapping $f$ of the unit ball $B^n$ of $\mathbb {C}^n$ into $\mathbb {C}^n$, we estimate the supremum of numbers $R$ such that some subdomain $\Omega$ of the ball is mapped by $f$ diffeomorphically onto some ball of radius $R$. Our estimates significantly improve earlier estimates, even in the case of harmonic functions in the disc.

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