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

Анотація

Let G be a simply connected domain in C which is T-homoheneous, i.e., TG = G for some T > 0. Let r(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 r(G). This implies a sharp lower bound for r(G) in terms of conformal modulus of a quadrilateral naturally connected with G