Hardy inequalities for simply connected planar domains

Abstract

Abstract. In 1986 A. Ancona showed, using the Koebe one-quarter Theorem, that for a simply-connected planar domain the constant in the Hardy inequality with the distance to the boundary is greater than or equal to 1/16. In this paper we consider classes of domains for which there is a stronger version of the Koebe Theorem. This implies better estimates for the constant appearing in the Hardy inequality. 1. Main result and discussion Let Ω be a domain in R2 and let Ωc = R2 \\ Ω be its complement. For any function u ∈ C 1 0 (Ω) we have: (1.1)