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Variation formulas for principal functions, II: Applications to variation for harmonic spans

Published online by Cambridge University Press:  11 January 2016

Sachiko Hamano ,
Fumio Maitani  and
Hiroshi Yamaguchi
Sachiko Hamano
Affiliation:
Department of Mathematics, Faculty of Human Development and Culture Fukushima University, Fukushima 960-1296, Japanhamano@educ.fukushima-u.ac.jp
Fumio Maitani
Affiliation:
2-7-7 Hiyoshidai, Ohtsu Shiga 522-0112, Japanhadleighbern@ybb.ne.jp
Hiroshi Yamaguchi
Affiliation:
2-6-20-3 Shiromachi, Hikone Shiga 522-0068, Japanh.yamaguchi@s2.dion.ne.jp
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Abstract

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A domain D ⊂ Cz admits the circular slit mapping P(z) for a, b ∈ D such that P(z) – 1/(z – a) is regular at a and P(b) = 0. We call p(z) = log|P(z)| the Li-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 = α – β 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 t ∈ B:= {t ∈ C: |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 = ∪t∈B(t, D (t)) is pseudoconvex in B × Cz, then s(t) is subharmonic on B. As a direct application, we have the subharmonicity of log cosh d(t) on B, where d(t) is the Poincaré distance between a and b on D(t).

Keywords

32T8530C25
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 204 , December 2011 , pp. 19 - 56
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2011

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