On the inequality $μ_3 < λ_1$ for the eigenvalues of the Laplace operator

Abstract: In 1952, Pólya proved the inequality $μ_2 &lt; λ_1$ for the eigenvalues of Dirichlet and Neumann problems in domain $Ω\subset \mathbb{R}<sup>d$, $d \ge 2$. We show that $μ_3 &lt; λ_1$ if $d\ge 3$. In the two-dimensional case, in 2025, Rohleder proved that $μ_3 &lt; λ_1$ for simply connected domains. We prove that this inequality holds also true for double connected domains symmetric with respect to the origin.