A sharp upper bound for the first Dirichlet eigenvalue of a class of wedge-like domains

By introducing new geometric factors which lend themselves to the Payne interpretation in Weinstein fractional space, we prove new isoperimetric inequalities which complement those of Payne–Weinberger and Saint-Venant giving a new upper bound for the fundamental mode of vibration of a wedge-like membrane and a new lower bound for its “relative torsional rigidity”. We also prove a new weighted version of a result of Crooke–Sperb for the associated fundamental eigenfunction of the Dirichlet Laplacian for such domains. A new weighted Rellich-type identity for wedge-like domains is also proved to achieve this latter task.