On the mean curvatures sharp estimates of hypersurfaces

Sharp estimates for the mean curvatures of hypersurfaces in Riemannian manifolds are known from the works of Jorge-Xavier [3], Markvorsen [6] and Vlachos [11]. We first give a simplified proof of these estimates. This proof shows that a similar original result holds for hypersurfaces in Einstein manifolds which are warped product of by Ricci-flat manifolds.