Optimal Curvature Inequalities for Partially Slant Submanifolds in Complex Space Forms

The class of partially slant submanifolds was introduced recently by Yerlikaya, Poyraz, and Şahin [Mediterr. J. Math. 22 (2025), 78] as a unifying framework for several important slant‐type geometries. In this paper, we establish sharp ‐Casorati inequalities for partially slant submanifolds of complex space forms. The obtained estimates relate the normalized scalar curvature to generalized ‐Casorati curvatures, the holomorphic sectional curvature of the ambient manifold, and geometric quantities naturally associated with the partially slant structure, namely the dimension and slant angle of the slant distribution, together with a geometric invariant measuring the interaction between the almost complex structure and the complementary distribution. The proof of the inequalities is based on a direct algebraic optimization method applied to the second fundamental form, in contrast with previous approaches relying on special optimization techniques on submanifolds. A complete pointwise characterization of the equality case is obtained, showing that equality holds precisely when all shape operators admit a common diagonal form with a distinguished tangent direction. Several consequences are derived for slant, semi‐slant, hemi‐slant, and bi‐slant submanifolds, and explicit examples are provided to illustrate both the sharpness and the strictness of the obtained inequalities.