2010/12/01 by S. Hamano, Hamano, S., F. Maitani +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Primary 32Txx #Secondary 30C25 #math.CV #msc:30C25 #msc:32Txx
paper · pdf · doi:10.48550/arxiv.1012.0208
30 pages, 4 figures, MSJ Autumn meeting in Nagoya(Japan) Sep. 20-23, 2010
arxiv created 2010/12/01 · openalex publication_date 2010/12/01 · arxiv updated 2010/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a domain D in ℂz with smooth boundary and for a,b∈ D, a≠ b, we have the circular (radial) slit mapping P(z)(Q(z)) on D such that P(z)- (1)/(z-a) (Q(z)- (1)/(z-a)) is regular at a and P(b)(Q(b))=0, and we call p(z)=log |P(z)| (q(z)=log|Q(z)|) the L1-(L0-)principal function; α=log|P'(b)| (β=log|Q'(b)|) the L1-(L0-)constant, and s=α- β the harmonic span, for D. S. Hamano in \citehamano-2 showed the variation formula of the second order for the L1-const. 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-const. β(t) for D(t), and combine these formulas to obtain, if the total space \mathcal D=∪t∈ B(t, D(t)) is pseudoconvex in B × ℂz, then s(t) is subharmonic on B. Since the geometric meaning of s(t) is showed, this fact gives one of the relations between the conformal mappings on each fiber D(t), t∈ B and the pseudoconvexity of \mathcal D. As a simple application we obtain the subharmonicity of log \cosh d(t) on B, where d(t) is the Poincaré distance between a and b.