An inequality for the zeta function of a planar domain

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.We prove non-negativeness and growth properties for $ζ\_Ω(s)-2\big({L(\partial Ω)\over 2π}\big)^sζ\_R(s)\ (s\leq-1)$, where $L(\partial Ω)$ is the length of the boundary curve and $ζ\_R$ stands for the classical Riemann zeta function.Two analogs of these results are also provided.

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