The fractional Makai-Hayman inequality

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