An inequality for the Steklov spectral zeta function of a planar domain

Abstract

We consider the zeta function ${\zeta }_{\Omega }$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $\Omega$ bounded by a smooth closed curve. We prove that, for a fixed real $s$ satisfying $|s|>1$ and fixed length $L(\partial \Omega )$ of the boundary curve, the zeta function ${\zeta }_{\Omega }(s)$ reaches its unique minimum when $\Omega$ is a disk.This result is obtained by studying the difference ${\zeta }_{\Omega }(s)-2(\frac{L(\partial \Omega )}{2\pi }{)}^s{\zeta }_R(s)$, where ${\zeta }_R$ stands for the classical Riemann zeta function. The difference turns out to be non-negative for real $s$ satisfying $|s|>1$. We prove some growth properties of the difference as $s\to \pm \infty$. Two analogs of these results are also provided.