An inequality for the Steklov spectral zeta function of a planar domain (Q1708571)

Summary: 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 that, for a fixed real \(s\) satisfying \(|s|>1\) and fixed length \(L(\partial\Omega)\) of the boundary curve, the zeta function \(\zeta_\Omega(s)\) reaches its unique minimum when \(\Omega\) is a disk. This result is obtained by studying the difference \(\zeta_\Omega(s)-2\left(\frac{L(\partial\Omega)}{2\pi}\right)^s\zeta_R(s)\), where \(\zeta_R\) stands for the classical Riemann zeta function. The difference turns out to be non-negative for real \(s\) satisfying \(|s|>1\). We prove some growth properties of the difference as \(s\to\pm\infty\). Two analogs of these results are also provided.