An Estimate for the Steklov Zeta Function of a Planar Domain Derived from a First Variation Formula

Abstract

We consider the Steklov zeta function \(\zeta _\varOmega \) of a smooth bounded simply connected planar domain \(\varOmega \subset {\mathbb {R}}^2\) of perimeter \(2\pi \). We provide a first variation formula for \(\zeta _\varOmega \) under a smooth deformation of the domain. On the base of the formula, we prove that, for every \(s\in (-1,0)\cup (0,1)\), the difference \(\zeta _\varOmega (s)-2\zeta _\mathrm{R}(s)\) is non-negative and is equal to zero if and only if \(\varOmega \) is a round disk (\(\zeta _\mathrm{R}\) is the classical Riemann zeta function). Our approach gives also an alternative proof of the inequality \(\zeta _\varOmega (s)-2\zeta _\mathrm{R}(s)\ge 0\) for \(s\in (-\infty ,-1]\cup (1,\infty )\); the latter fact was proved in our previous paper (2018) in a different way. We also provide an alternative proof of the equality \(\zeta '_\varOmega (0)=2\zeta '_\mathrm{R}(0)\) obtained by Edward and Wu (Determinant of the Neumann operator on smooth Jordan curves. Proc Am Math Soc 111(2):357–363, 1991).