A sharp isoperimetric bound for the ratio of the fifth and fourth eigenvalues of Dirichlet Laplacian

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a bounded domain with $n\ge 2$. The well-known Payne-Pólya-Weinberger conjecture for consecutive eigenvalue ratios asserts that $$\frac{{\lambda }_{m+1}(\mathrm{\Omega })}{{\lambda }_m(\mathrm{\Omega })}\le {\left(\frac{j_{n\mathrm{/}2,1}}{j_{n\mathrm{/}2-1,1}}\right)}^2,m=1,2,3,\cdots \mathrm{.}$$ Here, $j_{p,k}$ denotes the $k$-th positive zero of the Bessel function $J_p$, and ${\lambda }_j(\mathrm{\Omega })$ are the Dirichlet eigenvalues of $\mathrm{\Omega }$, counted with multiplicity and ordered as $$0<{\lambda }_1(\mathrm{\Omega })<{\lambda }_2(\mathrm{\Omega })\le {\lambda }_3(\mathrm{\Omega })\le \cdots \mathrm{.}$$ The cases $m=1,2,3$ were established by Ashbaugh and Benguria in their seminal works. In this paper, we give an affirmative answer to the Payne-Pólya-Weinberger conjecture for $m=4$ by combining a quadratic-form identity with a Brouwer degree argument. We also extend the corresponding spectral inequality to arbitrary bounded open sets in ${\mathbb{R}}^n$.