The objective of this paper is to achieve the inequality for Ricci curvature of a semi-slant warped product submanifold isometrically immersed in a generalized complex space form admitting a nearly Kaehler structure in the expressions of the squared norm of mean curvature vector and warping function. In addition, the equality case is likewise discussed. We provide numerous physical applications of the derived inequalities. Later, we proved that under a certain condition the base manifold $N_T^{n_1}$ is isometric to a $n_1$-dimensional sphere $S^{n_1}(\frac{{\lambda }_1}{n_1})$ with constant sectional curvature $\frac{{\lambda }_1}{n_1}.$

In this study, we seek to establish new upper bounds for the mean curvature and constant sectional curvature of the first positive eigenvalue of the $\alpha$-Laplacian operator on Riemannian manifolds. More precisely, various methods are used to determine the first eigenvalue for the $\alpha$-Laplacian operator on the closed oriented pseudo-slant submanifolds in a generalized Sasakian space form. From our findings for the Laplacian, we extend many Reilly-like inequalities to the $\alpha$-Laplacian on pseudo slant submanifold in a unit sphere.