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 ϕ-Laplacian associated with the generalized modified Ricci curvature, we first obtain sharp lower bound estimates for the first eigenvalue of the ϕ-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 : ∂M → RN(K) into space form RN(K).