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We consider the zeta function $\zeta_\Omega$ for the Dirichlet-to-Neumannoperator of a simply connected planar domain $\Omega$ bounded by a smoothclosed curve of perimeter $2\pi$. 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 proofthat for a fixed real $s$ satisfying $s\le-1$ the estimate$\zeta_\Omega''(s)\ge \zeta_{\mathbb{D}}''(s)$ holds with equality if and onlyif $\Omega$ is a disk. We then bring examples of domains $\Omega$ close to theunit disk where this estimate fails to be extended to the interval $(0,2)$.Other computations related to previous works are also detailed in the remainingpart of the text.

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