The fractional Makai–Hayman inequality

Abstract

AbstractWe prove that the first eigenvalue of the fractional Dirichlet–Laplacian of orderson a simply connected set of the plane can be bounded from below in terms of its inradius only. This is valid for$$1/2<s<1$$1/2<s<1and we show that this condition is sharp, i.e., for$$0<s\le 1/2$$0<s≤1/2such a lower bound is not possible. The constant appearing in the estimate has the correct asymptotic behavior with respect tos, as it permits to recover a classical result by Makai and Hayman in the limit$$s\nearrow 1$$s↗1. The paper is as self-contained as possible.