An inequality for the zeta function of a planar domain

Abstract

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.We prove non-negativeness and growth properties for $\zeta \_\Omega (s)-2(\frac{L(\partial \Omega )}{2\pi }{)}^s\zeta \_R(s)(s\le -1)$, where $L(\partial \Omega )$ is the length of the boundary curve and $\zeta \_R$ stands for the classical Riemann zeta function.Two analogs of these results are also provided.