Mean curvature of extrinsic spheres in submanifolds of real space forms

Given a submanifold P-m with the Hilbert-Schmidt norm of its second fundamental form bounded from above, in a real space form of constant curvature b, K-n (b), we have obtained a lower bound for the norm of the mean curvature normal vector field of extrinsic spheres with sufficiently small radius in P-m in terms of the mean curvature of the geodesic spheres in K-m (b), with same radius, and the mean curvature of P-m.