Geometric inequalities for CR b-invariant on generic statistical submanifolds

B.-Y. Chen [8] introduced the notion of CR b-invariant on CR-submanifolds. Recently, F.R. Al-Solamy et al. [3, 4] and I. Mihai et al. [11], respectively, established optimal inequalities for this invariant on anti-holomorphic submanifolds in complex space forms and for generic submanifolds in Sasakian space forms. Furthermore, A.N. Siddiqui et al. [19] derived equivalent inequalities for the contact CR b-invariant, but in the context of a generic submanifold within trans-Sasakian generalized Sasakian space forms. They also managed to identify a lower limit for the squared norm of the mean curvature. This was achieved by relating it to a CR b-invariant and the Laplacian of the warping function. This was done in the case of CR-warped products existing within the same ambient space forms. In the present paper, we obtain two optimal inequalities involving the CR b-invariant for a generic statistical submanifold in a holomorphic statistical manifold of constant holomorphic sectional curvature. Finally, we consider a generic statistical submersion from a holomorphic statistical manifold onto a statistical manifold.