The aim of this paper is to give a complementary upper bound-type isoperimetric inequality for the fundamental Dirichlet eigenvalue of a bounded domain completely contained in a cone. This inequality is a counterpart to the Ratzkin inequality for Euclidean wedge domains in higher dimensions. We also give a new version of the Crooke–Sperb inequality involving a new geometric quantity for the first eigenfunction of the Dirichlet Laplacian for such a class of domains.

中文翻译：

---

锥形域的第一个狄利克雷特征值的尖锐上限

本文的目的是给出完全包含在锥体中的有界域的基本 Dirichlet 特征值的互补上限型等周不等式。这种不等式与更高维的欧几里得楔形域的 Ratzkin 不等式相对应。我们还给出了 Crooke-Sperb 不等式的新版本，其中涉及此类域的 Dirichlet Laplacian 的第一本征函数的新几何量。