By using stochastic analysis on Riemannian manifold, we derive explicit constants c1(D) and
c2(D) for a d-dimensional compact Riemannian manifold D with boundary such thatholds for any Dirichlet eigenfunction φ on D with eigenvalue λ. In particular, when D is convex
with nonnegative Ricci curvature, this estimate holds for
holds for any Dirichlet eigenfunction φ on D with eigenvalue λ. In particular, when D is convex
with nonnegative Ricci curvature, this estimate holds for
