Zi-Kang Deng

A Proof of a Conjecture on a P\'olya Functional

Let $\lambda_1(\Omega)$ and $T(\Omega)$ denote the first Dirichlet eigenvalue and torsional rigidity of a bounded convex domain $\Omega\subset\mathbb{R}^2$, and let $M=\max_\Omega u$, where $u$ is the torsion function. We prove the sharp inequalities $\pi^2/24<\lambda_1(\Omega)T(\Omega)/|\Omega|<\pi^2/12$. This proves...