Four unknown constants∗

Let Ω ⊂ R2 be an arbitrary simply connected domain in the plane. We define RΩ = supz∈Ω dΩ(z) (the inradius of the domain) where dΩ(z) is the distance from z to the boundary of Ω. Let σΩ(z) be the density of the hyperbolic metric in Ω and let σΩ = infz∈Ω σΩ(z). Finally, denote by λ1 the lowest eigenvalue for the Dirichlet Laplacian in Ω and denote by τΩ the first exit time of Brownian motion from Ω. The following four inequalities hold. 1. There exists a positive constant C1, independent of the domain, such that for all functions u ∈ C∞ 0 (Ω)