Contact CR-Warped Product of Submanifolds of the Generalized Sasakian Space Forms Admitting a Nearly Trans-Sasakian Structure

In the present paper, we apply Hopf’s lemma to the contact CR-warped product submanifolds of generalized Sasakian space forms admitting nearly trans-Sasakian structure and establish a characterization inequality for the existence of these types of warped products. The indicated inequality generalizes the inequalities obtained in [M. Atceken, Bull. Iranian Math. Soc., 39, No. 3, 415–429 (2013), M. Atceken, Collect. Math., 62, No. 1, 17–26 (2011), and S. Sular and C. Özgür, Turkish J. Math., 36, 485–497 (2012)]. Moreover, we also deduce another inequality for the squared norm of the second fundamental form in terms of warping functions. This inequality is a generalization of the inequalities obtained in [I. Mihai, Geom. Dedicata, 109, 165–173 (2004) and K. Arslan, R. Ezentas, I. Mihai, and C. Murathan, J. Korean Math. Soc., 42, No. 5, 1101–1110 (2005)]. The inequalities deduced in the paper either generalize or improve all inequalities available in the literature and related to the squared norm of the second fundamental form for the contact CR-warped product submanifolds of any almost contact metric manifold.