Abstract: Given a compact Riemannian manifold \((M^n,g)\) with boundary \(\partial M\), we give an estimate for the quotient \(\displaystyle{\frac{\int_{\partial M} fd\mu_g}{\int_M fd\mu_g}}\), where \(f\) is a smooth positive function defined on \(M\) that satisfies some inequality involving the scalar Laplacian. By the mean value lemma established by Alessandro Savo in A mean value lemma and applications, Bull. Soc. Math. France 129 (2001), 505-542, 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.