Convexity properties of the difference over the real axis between the Steklov zeta functions of a smooth planar domain with $2\pi$ perimeter and of the unit disk

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 of perimeter $2\pi$. We prove that ${\zeta }_{\Omega }^{\prime \prime }(0)\ge {\zeta }_{\mathbb{D}}^{\prime \prime }(0)$ with equality if and only if $\Omega$ is a disk where $\mathbb{D}$ denotes the closed unit disk. We also provide an elementary proof that for a fixed real $s$ satisfying $s\le -1$ the estimate ${\zeta }_{\Omega }^{\prime \prime }(s)\ge {\zeta }_{\mathbb{D}}^{\prime \prime }(s)$ holds with equality if and only if $\Omega$ is a disk. We then bring examples of domains $\Omega$ close to the unit disk where this estimate fails to be extended to the interval $(0,2)$. Other computations related to previous works are also detailed in the remaining part of the text.