A Faber--Krahn inequality for indented and cut membranes

In 1960, Payne and Weinberger proved that among all domains that lie within a wedge (an angle whose measure is less than or equal to $\pi$), and have a given value of a certain integral the circular sector has the lowest fundamental eigenvalue of the Dirichlet Laplacian. Here, it is shown that an analogue of this assertion is true for domains with a cut and for indented domains; that is, for those located in a reflex angle (its measure is between $\pi$ and $2 \pi$).