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

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 $2pi$. We prove that $zeta_Omega(0)ge zeta_{mathbb{D}}(0)$ 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 $sle-1$ the estimate $zeta_Omega(s)ge zeta_{mathbb{D}}(s)$ 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.