Contact CR $\delta$-invariant: an optimal estimate for Sasakian statistical manifolds

Chen (1993) developed the theory of $\delta$-invariants to establish novel necessary conditions for a Riemannian manifold to allow a minimal isometric immersion into Euclidean space. Later, Siddiqui et al. (2024) derived optimal inequalities involving the CR $\delta$-invariant for a generic statistical submanifold in a holomorphic statistical manifold of constant holomorphic sectional curvature. In this work, we extend the study of such optimal inequality to the contact CR $\delta$-invariant on contact CR-submanifolds in Sasakian statistical manifolds of constant $\varphi$-sectional curvature. This paper concludes with a summary and final remarks.