Abstract
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.