We obtain sharp estimates involving the mean curvatures of higher order of a complete bounded hypersurface immersed in a complete Riemannian manifold. Similar results are also given for complete spacelike hypersurfaces in Lorentzian ambient spaces.Estimates for the k-mean curvatures H k of higher order of a compact hypersurface in a complete Riemannian manifold have been subsequently obtained by Vlachos [14], Veeravalli [13], Fontenele-Silva [9], Roth [12] and Ranjbar-Motlagh [11]. In this paper, we generalize a result given in the latter that we describe next.Let f : M n →M n+1 be a codimension one isometric immersion between complete Riemannian manifolds. Assume that the hypersurface lies inside a closed geodesic ball BM (r) of radius r and center o ∈M n+1 and that 0 < r < min{injM (o), π/2 √ b} where injM (o) is the injectivity radius at o and π/2 √ b is replaced by +∞ if b ≤ 0. Suppose also that there is a point p 0 ∈ M n such that f (p 0 ) ∈ SM (r) where SM (r) is the boundary of BM (r). In the context of this paper, this is a slightly weaker assumption than asking M n to be compact. Let K rad M denote the radial sectional curvatures in BM (r) along geodesics issuing from the center and assume that K