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Variation formulas for principal functions. II: Applications to variation for harmonic spans. (English) Zbl 1234.32008

Summary: A domain \(D\subset\mathbb{C}_{z}\) admits a circular slit mapping \(P(z)\) for \(a,b\in D\) such that \(P(z)-1/(z-a)\) is regular at \(a\) and \(P(b)=0\). We call \(p(z)=\log|P(z)|\) the \(L_{1}\)-principal function and \(\alpha =\log|P'(b)|\) the \(L_{1}\)-constant, and similarly, the radial slit mapping \(Q(z)\) implies a \(L_{0}\)-principal function \(q(z)\) and a \(L_{0}\)-constant \(\beta\). We call \(s=\alpha-\beta \) the harmonic span for \((D,a,b)\). We show the geometric meaning of \(s\). S. Hamano [Mich. Math. J. 60, No. 2, 271–288 (2011; Zbl 1235.30028)] showed a variation formula for the \(L_{1}\)-constant \(\alpha(t)\) for the moving domain \(D(t)\) in \(\mathbb{C}_{z}\) with \(t\in B:=\{t\in \mathbb{C}:|t|<\rho\}\). We show a corresponding formula for the \(L_{0}\)-constant \(\beta(t)\) for \(D(t)\) and combine these to prove that, if the total space \(\mathcal{D}=\bigcup_{t\in B}(t,D(t))\) is pseudoconvex in \(B\times\mathbb{C}_{z}\), then \(s(t)\) is subharmonic on \(B\). As a direct application, we get the subharmonicity of \(\log \cosh d(t)\) on \(B\), where \(d(t)\) is the Poincaré distance between \(a\) and \(b\) on \(D(t)\).

MSC:

32T99 Pseudoconvex domains
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions

Citations:

Zbl 1235.30028
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References:

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