2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/112757In [6] we proved Chen's inequality regarded as a problem of constrained maximum. In this paper we introduce a Riemannian invariant obtained from Chen's invariant, replacing the sectional curvature by the Ricci curvature of k-order. This invariant can be estimated, in the case of submanifolds M in space forms $\widetilde{M}(c)$, varying with c and the mean curvature of M in $\widetilde{M}(c)$. A improuvement of this inequality in the Lagrangian case is obtained.The paper is submitted to Rocky Mountain Journal of MathematicsDifferential GeometryOptimization and Control53C21, 53C24, 49K35text