Jorge-Koutrofiotis and Pigola-Rigoli-Setti proved sharp sectional curvature estimates for extrinsically bounded submanifolds. Alias, Bessa and Montenegro showed that these estimates hold on properly immersed cylindrically bounded submanifolds. On the
other hand, Alias, Bessa and Dajczer proved sharp mean curvature estimates for properly immersed cylindrically bounded submanifolds. In this paper we prove these sectional and mean curvature estimates for a larger class of submanifolds, the properly immersed $phi$-bounded submanifolds.
Based on ideas of L. Alias, D. Impera and M. Rigoli developed in Hypersurfaces of constant higher order mean curvature in warped products, we develope a fairly general weak/Omori-Yau maximum principle for trace operators. We apply this version of max
imum principle to generalize several higher order mean curvature estimates and to give an extension of Alias-Impera-Rigoli Slice Theorem
