Generalized inequalities on warped product submanifolds in nearly trans-Sasakian manifolds

In N.this paper, we study warped product submanifolds of nearly trans-Sasakian manifolds. The non-existence of warped product semi-slant submanifolds of type N-theta x(f)N(T) is shown, whereas some characterization and new geometric obstructions are obtained for the warped products of type N-T x(f) N-theta. We establish two general inequalities for the squared norm of the second fundamental form. The first inequality generalizes derived inequalities for some contact metric manifolds (Kadri et al. in J. Korean Math. Soc. 42: 1101-1110, 2005; Munteanu in Publ. Math. (Debr.) 66: 75-120, 2005; Mustafa et al. in Taiwan. J. Math. 17: 1473-1486, 2013; Uddin and Khan in J. Inequal. Appl. 2012: 304, 2012), while by a new technique, the second inequality is constructed to express the relation between extrinsic invariant (second fundamental form) and intrinsic invariant (scalar curvatures). The equality cases are also discussed.