Some geometric inequalities on Riemannian manifolds associated with the generalized modified Ricci curvature

In this paper, by establishing a Reilly-type formula with respect to the phi-Laplacian associated with the generalized modified Ricci curvature, we first obtain sharp lower bound estimates for the first eigenvalue of the phi-Laplacian. On the other hand, some new Brascamp-Lieb-type and Colesanti-type inequalities under some suitable boundary conditions are achieved. As applications, we also obtain some relationships between the weighted mean curvature of a boundary submanifold and the mean curvature of submanifold x : partial derivative M ->(RK)-K-N() into space form R-N(K).