Neumann--Dirichlet eigenvalue comparison at the first Dirichlet threshold in the plane

Let $\mathrm{\Omega }\subset {\mathbb{R}}^2$ be a bounded connected Lipschitz domain, and let $\{{\mu }_j(\mathrm{\Omega }){\}}_{j\ge 1}$ and $\{{\lambda }_j(\mathrm{\Omega }){\}}_{j\ge 1}$ denote the Neumann and Dirichlet Laplacian eigenvalues, respectively, counted with multiplicity. We prove that $${\mu }_3(\mathrm{\Omega })<{\lambda }_1(\mathrm{\Omega }),$$ thereby removing the simple-connectivity assumption from the previously known planar result at the first Dirichlet threshold. We also establish a three-spectrum counting inequality relating the Dirichlet, Neumann, and conductivity spectra.