Chen-Type Inequalities for PS-Submanifolds in Complex Space Forms

In this paper, we investigate Chen’s δ-invariant for partially slant (PS) submanifolds of complex space forms. A PS-submanifold admits an orthogonal decomposition of the tangent bundle into a proper slant distribution and an arbitrary ambiguous distribution. Using the Gauss equation together with algebraic optimization techniques, we derive a Chen-type inequality relating the δ-invariant to the squared mean curvature, the holomorphic sectional curvature of the ambient space, and the slant angle of the slant distribution. Unlike the classical Chen inequality for slant submanifolds, the obtained estimate contains an additional term reflecting the contribution of the ambiguous distribution. Several corollaries are derived, including dimension-dependent bounds and special cases corresponding to hemi-slant and semi-slant submanifolds. The equality case is also characterized in terms of the structure of the shape operators. These results provide a natural extension of Chen-type inequalities to the broader framework of partially slant geometry in Kähler manifolds.