Recently, many authors studied the relations between the squared norm of the second fundamental form (extrinsic invariant) and the warping function (intrinsic invariant) for warped product submanifolds (see [1, 7, 14]). Inspired by those relations we establish a general sharp inequality, namely $\|h\|^2 \geq 2s[\|\nabla lnf\|^2+\alpha ^2 -\beta^2]$, for contact CR-warped products of nearly trans-Sasakian manifolds. Our inequality generalizes all derived inequalities for contact CR-warped products either in any contact metric manifold. The equality case is also handled.
"CONTACT CR-WARPED PRODUCT SUBMANIFOLDS OF NEARLY TRANS-SASAKIAN MANIFOLDS." Taiwanese J. Math. 17 (4) 1473 - 1486, 2013. https://doi.org/10.11650/tjm.17.2013.2601