**1. Introduction**

The problem of discovering simple relationships between the main *intrinsic invariants* and the main *extrinsic invariants* of submanifolds is a basic problem in submanifold theory [1]. In this respect, beautiful results focus on certain types of geometric inequalities. Moreover, another basic problem in this field is to study the *ideal submanifolds* in a space form, namely to investigate the submanifolds which satisfy the equality case of such inequalities [2].

The method of looking for *Chen invariants* answers the problems posed above. First, Chen demonstrated in [3] an optimal inequality for a submanifold on a real space form between the intrinsically defined *δ*-curvature and the extrinsically defined squared mean curvature. This approach initiated a new line of research and was extended to various types of submanifolds in several types of ambient spaces, e.g., submanifolds in complex space forms of constant holomorphic sectional curvature (see [4–7]). The submanifolds attaining the equality of these inequalities (called Chen ideal submanifolds) were also investigated. Recently, Chen et al. classified *δ*(2, *n* − 2)-ideal Lagrangian submanifolds in complex space forms in [8].

Moreover, new solutions to the above problems are given by the inequalities involving *δ-Casorati curvatures*, initiated in [9,10]. In the search for a true measure of curvature, Casorati in 1890 proposed the curvature which nowadays bears his name because it better corresponds with our *common intuition of curvature* than Gauss and mean curvature [11]. However, this notion of curvature was soon forgotten and was rediscovered by Koenderink working in the field of computer vision [12]. Verstraelen developed some geometrical models for early vision, presenting perception via the Casorati curvature of sensation [13]. A geometrical interpretation of this type of curvature for submanifolds in Riemannian spaces was given in [14]. In [15], the isotropical Casorati curvature of production surfaces

was studied. The Casorati curvature was used to obtain optimal inequalities between intrinsic and extrinsic curvatures of submanifolds in real space forms in [9,10]. Later, this knowledge was extended (e.g., see [16–21]). Submanifolds which satisfy these equalities are named *Casorati ideal submanifolds*. Recently, Vîlcu established an optimal inequality for Lagrangian submanifolds in complex space forms involving Casorati curvature [22]. Aquib et al. obtained a classification of Casorati ideal Lagrangian submanifolds in complex space forms [23]. Very recently, Suceav˘a and Vajiac studied inequalities involving some Chen invariants, mean curvature, and Casorati curvature for strictly convex Euclidean hypersurfaces [24]. Brubaker and Suceav ˘a investigated a geometric interpretation of Cauchy–Schwarz inequality in terms of Casorati curvature [25].

The concept of *statistical manifold* was defined by Amari in 1985, in the basic study on information geometry [26]. Currently, interest in the field of statistical manifolds is increasing, being focused on applications in differential geometry, information geometry, statistics, machine learning, etc. (see, e.g., [27–29]). Cuingnet et al. introduced a continuous framework to spatially regularize support vector machines (SVM) for brain image analysis, considering the images as elements of a statistical manifold, in order to classify patients with Alzheimer's disease [30]. The study of curvature invariants of submanifolds in statistical manifolds gives other solutions to the above research problems. Aydin et al. established some inequalities (Chen–Ricci and Wintgen) for submanifolds in statistical manifolds of constant curvature in [31,32]. Lee et al. obtained inequalities on Sasakian statistical manifolds in terms of Casorati curvatures [33]. Aquib and Shahid [34] proved some inequalities involving Casorati curvatures on statistical submanifolds in quaternion Kähler-like statistical space forms. The quaternionic theory of statistical manifolds is investigated in [35]. Very recently, new results have been published. Aytimur et al. established some Chen inequalities for submanifolds in Kähler-like statistical manifolds [36]. Aquib et al. achieved generalized Wintgen-type inequalities for submanifolds in generalized space forms [37]. Chen et al. established a Chen first inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature [38]. Moreover, Siddiqui et al. studied a Chen inequality for statistical warped products statistically immersed in a statistical manifold of constant curvature [39].

Recently, Furuhata et al. [40] defined the notion of a *holomorphic statistical manifold*, which can be considered as a generalization of a special Kähler manifold. The authors establish the basics for statistical submanifolds in holomorphic statistical manifolds.

In order to find out new solutions for the problems under debate, we obtain inequalities for statistical submanifolds in holomorphic statistical manifolds. The invariants involved in such inequalities are the extrinsic normalized *δ*-Casorati curvatures and the intrinsic scalar curvature. The method is focused on a constrained extremum problem. Moreover, the equality cases are investigated. This study revealed that the equality at all points characterizes submanifolds that are totally geodesic with respect to the Levi–Civita connection.
