Ashbaugh–Benguria reciprocal-gap conjecture for Dirichlet eigenvalues

Let $N\ge2$, let $\Omega\subset\mathbb R^N$ be a bounded domain, and let $\nu=(N-2)/2$. Denote the Dirichlet eigenvalues of $\Omega$ by $0<\lambda_1(\Omega)<\lambda_2(\Omega)\le\lambda_3(\Omega)\le\cdots$, and let $j_{\mu,k}$ be the $k$th positive zero of the Bessel function $J_\mu$. Ashbaugh–Benguria's reciprocal-gap conjecture. Every such domain satisfies

$$\sum_{i=1}^{N} \frac{\lambda_1(\Omega)}{\lambda_{i+1}(\Omega)-\lambda_1(\Omega)} \ge \frac{N}{j_{\nu+1,1}^2/j_{\nu,1}^2-1},$$

with equality if and only if $\Omega$ is a Euclidean ball. The conjecture was stated as a sharp reciprocal-gap inequality in the Payne–Pólya–Weinberger program and was later listed among open problems; the present paper proves the inequality in every dimension and gives the sharper capacitary equality characterization.