A sharp inequality for the order of the minimal positive harmonic function in $T$-homogeneous domain

V. Azarin, A. Gol'dberg

Department of Mathematics, Bar-Ilan University, Ramat-Gan, 52900, Israel

Abstract: Let $G$ be a simply connected domain in $\mathbb C$ which is $T$-homoheneous, i.e., $TG=G$ for some $T>0$. Let $\rho(G)$ be the order of the minimal positive harmonic function in $G$. We prove that a kind of symmetrization of $G$ and prove that it does not increase $\rho(G)$. This implies a sharp lower bound for $\rho(G)$ in terms of conformal modulus of a quadrilateral naturally connected with $G$.

MSC: 31A05, 30C75

Received: 02.02.2004

Language: English

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