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December 2011 Variation formulas for principal functions, II: Applications to variation for harmonic spans
Sachiko Hamano, Fumio Maitani, Hiroshi Yamaguchi
Nagoya Math. J. 204: 19-56 (December 2011). DOI: 10.1215/00277630-1431822

Abstract

A domain DCz admits the circular slit mapping P(z) for a,bD such that P(z)1/(za) is regular at a and P(b)=0. We call p(z)=log|P(z)| the L1-principal function and α=log|P'(b)| the L1-constant, and similarly, the radial slit mapping Q(z) implies the L0-principal function q(z) and the L0-constant β. We call s=αβ the harmonic span for (D,a,b). We show the geometric meaning of s. Hamano showed the variation formula for the L1-constant α(t) for the moving domain D(t) in Cz with tB:={tC:|t|<ρ}. We show the corresponding formula for the L0-constant β(t) for D(t) and combine these to prove that, if the total space D=tB(t,D(t)) is pseudoconvex in B×Cz, then s(t) is subharmonic on B. As a direct application, we have the subharmonicity of logcoshd(t) on B, where d(t) is the Poincaré distance between a and b on D(t).

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Sachiko Hamano. Fumio Maitani. Hiroshi Yamaguchi. "Variation formulas for principal functions, II: Applications to variation for harmonic spans." Nagoya Math. J. 204 19 - 56, December 2011. https://doi.org/10.1215/00277630-1431822

Information

Published: December 2011
First available in Project Euclid: 5 December 2011

zbMATH: 1234.32008
MathSciNet: MR2863364
Digital Object Identifier: 10.1215/00277630-1431822

Subjects:
Primary: 32T85
Secondary: 30C25

Rights: Copyright © 2011 Editorial Board, Nagoya Mathematical Journal

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Vol.204 • December 2011
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