Calabi type functionals for coupled K\"ahler-Einstein metrics

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.