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

We consider the zeta function $ζ_Ω$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $Ω$ bounded by a smooth closed curve of perimeter $2π$. We prove that $ζ_Ω''(0)\ge ζ_{\mathbb{D}}''(0)$ with equality if and only if $Ω$ 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 $ζ_Ω''(s)\ge ζ_{\mathbb{D}}''(s)$ holds with equality if and only if $Ω$ is a disk. We then bring examples of domains $Ω$ 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.

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