An optimal lower bound in fractional spectral geometry for planar sets with topological constraints

We prove a lower bound on the first eigenvalue of the fractional Dirichlet–Laplacian of order (Formula presented.) on planar open sets, in terms of their inradius and topology. The result is optimal, in many respects. In particular, we recover a classical result proved independently by Croke, Osserman, and Taylor, in the limit as (Formula presented.) goes to 1. The limit as (Formula presented.) goes to (Formula presented.) is carefully analyzed, as well.