An optimal inequality for warped product submanifolds in complex space forms

In this work, using optimization procedures on Riemannian submanifolds, we obtained two different inequalities on the generalized normalized $ \delta $-Casorati curvatures of warped product submanifolds in complex space forms. We also quantified the conditions under which these inequalities become equalities, providing more insight into their geometric consequences. Further, we described new findings in the form of harmonic functions and Hessian functions, which offer a more general view of the interplay between curvature and analyticity.