We consider the zeta function ζΩ\zeta_{\Omega} for the Dirichlet-to-Neumann operator of a simply connected planar domain Ω bounded by a smooth closed curve of perimeter 2⁢π2\pi. We name the difference ζΩ-ζD\zeta_{\Omega}-\zeta_{\mathbb{D}} the normalized Steklov zeta function of the domain Ω, where 𝔻 denotes the closed unit disk. We prove that (ζΩ-ζD)′′⁢(0)≥0(\zeta_{\Omega}-\zeta_{\mathbb{D}})^{\prime\prime}(0)\geq 0 with equality if and only if Ω is a disk. We also provide an elementary proof that, for a fixed real 𝑠 satisfying s≤-1s\leq-1, the estimate (ζΩ-ζD)′′⁢(s)≥0(\zeta_{\Omega}-\zeta_{\mathbb{D}})^{\prime\prime}(s)\geq 0 holds with equality if and only if Ω is a disk. We then bring examples of domains Ω close to the unit disk where this estimate fails to be extended to the interval (0,2)(0,2). Other computations related to previous works are also detailed in the remaining part of the text.

中文翻译：

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平面域的归一化 Steklov zeta 函数的凸性特性

我们考虑由周长为 2⁢π2\pi 的平滑闭合曲线包围的单连通平面域 Ω 的 Dirichlet-to-Neumann 算子的 zeta 函数 ζΩ\zeta_{\Omega}。我们将差值 ζΩ-ζD\zeta_{\Omega}-\zeta_{\mathbb{D}} 命名为域 Ω 的归一化 Steklov zeta 函数，其中 𝔻 表示封闭的单位圆盘。我们证明 (ζΩ-ζD)′′⁢(0)≥0(\zeta_{\Omega}-\zeta_{\mathbb{D}})^{\prime\prime}(0)\geq 0 等式如果并且仅当 Ω 是圆盘时。我们还提供了一个基本证明，对于满足 s≤-1s\leq-1 的固定实数 𝑠，估计 (ζΩ-ζD)′′⁢(s)≥0(\zeta_{\Omega}-\zeta_{\ mathbb{D}})^{\prime\prime}(s)\geq 0 当且仅当 Ω 是圆盘时成立。然后，我们将域 Ω 的示例带到接近单位圆盘的位置，其中该估计无法扩展到区间 (0,2)(0,2)。