An estimate for the Steklov zeta function of a planar domain derived from a first variation formula

We consider the Steklov zeta function $zeta$ $Omega$ of a smooth bounded simply connected planar domain $Omega$ $subset$ R 2 of perimeter 2$pi$. We provide a first variation formula for $zeta$ $Omega$ 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 $zeta$ $Omega$ (s) -- 2$zeta$ R (s) is non-negative and is equal to zero if and only if $Omega$ is a round disk ($zeta$ R is the classical Riemann zeta function). Our approach gives also an alternative proof of the inequality $zeta$ $Omega$ (s) -- 2$zeta$ 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 $zeta$ $Omega$ (0) = 2$zeta$ R (0) obtained by Edward and Wu [1991].