Some Inequalities On Lagrangian Submanifolds In Holomorphic Statistical Manifolds Of Constant Holomorphic Sectional Curvature

The study of the relationships between the intrinsic invariants and the extrinsic invariants has always been a fundamental problem in the theory of geometry of submanifolds.Such relationships discovered until now arc mainly embodied in inequalities.On the other hand,holomor-phic statistical manifolds of constant holomorphic sectional curvatures can be viewed as a generalization of complex space forms.The main purpose of this paper is to establish some geometric inequalities be-tween the intrinsic invariants and the extrinsic invariants for Lagrange submanifolds of a holomorphic statistical manifold of constant holomor-phic sectional curvature.Specially speaking,we obtain the inequalities between the ?-Casorati curvature and the normalized scalar curvature.We also establish the DDVV-inequalities involving the normalized s-calar curvature and the normalized normal scalar curvature,and the equality cases have been discussed.In addition,we establish inequali-ties between the Oprea-invariant and the mean curvature,and the in-equality between the Chen-invariant and the mean curvature,which generalize the respective results for Lagrange submanfolds of complex space forms.