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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
##### Keywords:
pseudoconvex domain; subharmonic function
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##### References:
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