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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:
  L.V. Ahlfors and L. Sario, Riemann Surfaces , Princeton Math. Ser. 26 , Princeton University Press, Princeton, 1960. · Zbl 0196.33801  E. Bedford and B. Gaveau, Envelopes of holomorphy of certain 2-spheres in \Bbb C 2 , Amer. J. Math. 105 (1983), 975-1009. · Zbl 0535.32008 · doi:10.2307/2374301  H. Behnke, Die Kanten singuärer Mannigfaltigkeiten , Abh. Math. Semin. Univ. Hambg. 4 (1926), 347-365. · JFM 52.0343.01  M. Brunella, Subharmonic variation of the leafwise Poincaré metric , Invent. Math. 152 (2003), 119-148. · Zbl 1029.32014 · doi:10.1007/s00222-002-0269-0  L. Ford, Automorphic Functions , 2nd ed., Chelsea Publishing, New York, 1951. · JFM 55.0810.04  H. Grunsky, Neue Abschätzungen zur konformen Abbildung ein- und mehrfach zusammenhängender Beriche , Schr. Sem. Univ. Berlin 1 (1932), 95-140. · Zbl 0005.36204 · eudml:204228  R. Gunning and R. Narasimhan, Immersion of open Riemann surfaces , Math. Ann. 174 (1967), 103-108. · Zbl 0179.11402 · doi:10.1007/BF01360812 · eudml:161628  S. Hamano, A lemma on C 1 subharmonicity of the harmonic spans for the discontinuously moving Riemann surfaces , preprint to appear in J. Math. Soc. Japan. · Zbl 1294.30051  S. Hamano, Variation formulas for L 1 -principal functions and application to simultaneous uniformization problem , Michigan Math. J. 60 (2011), 271-288. · Zbl 1235.30028  S. Hamano, Variation formulas for principal functions, III: Applications to variation for Schiffer spans , · Zbl 1234.32008  N. Levenberg and H. Yamaguchi, The metric induced by the Robin function , Mem. Amer. Math. Soc. 448 (1991), 1-155. · Zbl 0742.31003  F. Maitani and H. Yamaguchi, Variation of Bergman metrics on Riemann surfaces , Math. Ann. 330 (2004), 477-489. · Zbl 1077.32006 · doi:10.1007/s00208-004-0556-8  M. Nakai and L. Sario, Classification Theory of Riemann Surfaces , Grundlehren Math. Wiss. 164 , Springer, New York, 1970. · Zbl 0199.40603  Y. Nishimura, Immersion analytique d’une famille de surfaces de Riemann ouverts , Publ. Res. Inst. Math. Sci. 14 (1978), 643-654. · Zbl 0434.32021 · doi:10.2977/prims/1195188831  T. Nishino, Function Theory in Several Complex Variables , Transl. Math. Monogr. 193 , Amer. Math. Soc., Providence, 2001. · Zbl 0972.32001  M. Schiffer, The span of multiply connected domains , Duke Math. J. 10 (1943), 209-216. · Zbl 0060.23704 · doi:10.1215/S0012-7094-43-01019-1  H. Yamaguchi, Variations of pseudoconvex domains over \Bbb C n , Michigan Math. J. 36 (1989), 415-457. · Zbl 0692.31004 · doi:10.1307/mmj/1029004011
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