An inequality for the zeta function of a planar domain

TL;DR

This paper establishes inequalities and growth properties for the zeta function associated with the Dirichlet-to-Neumann operator of a planar domain, linking geometric boundary length with classical zeta functions.

Contribution

It introduces new inequalities for the zeta function of a planar domain's Dirichlet-to-Neumann operator, connecting spectral properties with geometric boundary length.

Findings

Proved non-negativeness of a specific zeta function difference for s ≤ -1.

Established growth properties of the zeta function related to boundary length.

Provided two analogs of the main results.

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 $\zeta\_\Omega(s)-2\big({L(\partial \Omega)\over 2\pi}\big)^s\zeta\_R(s)\ (s\leq-1)$, 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.