  We introduce the coupled Ricci-Calabi functional and the coupled H-functional
which measure how far from a coupled K\"ahler-Einstein metric in the sense of
Hultgren-Witt Nystr\"om. We first give corresponding moment weight type
inequalities which estimate each functional in terms of algebraic invariants.
Secondly, we give corresponding Hessian formulas for these functionals at each
critical point, which have an application to a Matsushima type obstruction
theorem for the existence of a coupled K\"ahler-Einstein metric.
