  Given a compact Riemannian manifold (M n , g) with boundary $\partial$M , we
give an estimate for the quotient $\partial$M f d$\mu$ g M f d$\mu$ g , where f
is a smooth positive function defined on M that satisfies some inequality
involving the scalar Laplacian. By the mean value lemma established in [37], we
provide a differential inequality for f which, under some curvature
assumptions, can be interpreted in terms of Bessel functions. As an application
of our main result, a direct proof is given of the Faber-Krahn inequalities for
Dirichlet and Robin Laplacian. Also, a new estimate is established for the
eigenvalues of the Dirac operator that involves a positive root of Bessel
function besides the scalar curvature. Independently, we extend the Robin
Laplacian on functions to differential forms. We prove that this natural
extension defines a self-adjoint and elliptic operator whose spectrum is
discrete and consists of positive real eigenvalues. In particular, we
characterize its first eigenvalue and provide a lower bound of it in terms of
Bessel functions.
