Comparison Theorems for Manifold with Mean Convex Boundary

Let $M^n$ be an $n$-dimensional Riemannian manifold with boundary $\partial M$\mathrm{.}AssumethatRiccicurvatureisboundedfrombelowby$(n-1)k$,for$k\in \RR$,wegiveasharpestimateoftheupperboundof$\rho(x)=\dis(x, \partial M)$,intermsofthemeancurvatureboundoftheboundary\mathrm{.}When$\partial M$ is compact, the upper bound is achieved if and only if $M$ is isometric to a disk in space form. A Kaehler version of estimation is also proved. Moreover we prove a Laplace comparison theorem for distance function to the boundary of Kaehler manifold and also estimate the first eigenvalue of the real Laplacian.