A sharp bound for the first Robin-Dirichlet eigenvalue

In this paper, we study the first eigenvalue of the Laplacian on doubly connected domains when Robin and Dirichlet conditions are imposed on the outer and the inner part of the boundary, respectively. We provide that the spherical shell reaches the maximum of the first eigenvalue of this problem among the domains with fixed measure, outer perimeter and inner $(n-1{)}^{th}$ quermassintegral.