  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.
