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.