Reilly type inequality for the first eigenvalue of the Lr;F operator

Given a positive function F on Sn which satisfies a convexity condition, for 1⩽r⩽n, we define for hypersurfaces in Rn+1 the r-th anisotropic mean curvature function Hr;F, a generalization of the usual r-th mean curvature function. We also define Lr;F operator, the linearized operator of the (r+1)-th anisotropic mean curvature, which is a generalization of the usual Lr operator for hypersurfaces in the Euclidean space Rn+1. The Reilly type inequalities for the first eigenvalue of the Lr;F operator have been proved.