Cauchy–Schwarz-type inequalities on Kähler manifolds-II

We establish in this note some Cauchy–Schwarz-type inequalities on compact Kähler manifolds, which generalize the classical Khovanskii–Teissier inequalities to higher-dimensional cases. Our proof is to make full use of the mixed Hodge–Riemann bilinear relations due to Dinh and Nguyên. A proportionality problem related to our main result is also proposed.