  In the theory of submanifolds, the following problem is fundamental: to
establish simple relationships between the main intrinsic invariants and the
main extrinsic invariants of the submanifolds.The basic relationships
discovered until now [1, 2, 3, 4] are inequalities. To analyze these problems,
we follow the idea of C. Udriste [6, 7, 8] that the method of constrained
extremum is a natural way to prove geometric inequalities. We improve the
Chen's inequality which characterizes a totally real submanifold of a complex
space form. For that we suppose that the submanifold is Lagrangian and we
formulate and analyze a suitable constrained extremum problem.
