Mixed volume forms and a complex equation of Monge–Ampère type on Kähler manifolds of positive curvature

We consider a generalization of the Calabi problem. In the analytic set-up on a Kähler manifold, it leads to a complex Monge–Ampère equation containing the mixed discriminant of the given and unknown metrics. We obtain sufficient conditions for its solubility in the case when the Kähler manifold is $\delta$-pinched ($\delta>1/2$).