On the inequality ${\mu }_3<{\lambda }_1$ for the eigenvalues of the Laplace operator

In 1952, Pólya proved the inequality ${\mu }_2<{\lambda }_1$ for the eigenvalues of Dirichlet and Neumann problems in domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$, $d\ge 2$. We show that ${\mu }_3<{\lambda }_1$ if $d\ge 3$. In the two-dimensional case, in 2025, Rohleder proved that ${\mu }_3<{\lambda }_1$ for simply connected domains. We prove that this inequality holds also true for double connected domains symmetric with respect to the origin.