An inequality for the zeta function of a planar domain

We consider the zeta function $\zeta \mathrm{\_}\mathrm{\Omega }$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $\mathrm{\Omega }$ bounded by a smooth closed curve.We prove non-negativeness and growth properties for $\zeta_\Omega(s)-2\big({L(\partial \Omega)\over 2\pi}\big)^s\zeta_R(s)
(s\leq-1)$,where$L(\partial \Omega)$isthelengthoftheboundarycurveand$\zeta_R$ stands for the classical Riemann zeta function.Two analogs of these results are also provided.