Bounding the integral of powered $i$-th mean curvatures

We get estimates for the integrals of powered $i$-th mean curvatures, $1\leq i\leq n-1$, of compact and convex hypersurfaces, in terms of the quermaß integrals of the corresponding $C^2_+$ convex bodies. These bounds will be obtained as consequences of a most general result for functions defined on a general probability space. From this result, similar estimates for the integrals of any convex transformation of the elementary symmetric functions of the radii of curvature of $C^2_+$ convex bodies will be also proved, both, in terms of the quermaß integrals, and of the roots of their Steiner polynomials. Finally, the radial function is considered, and estimates of the corresponding integrals are obtained in terms of the dual quermaß integrals.