  We establish in this note some Cauchy-Schwarz-type inequalities on compact
K\"{a}hler 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 Nguy$\hat{\text{e}}$n. A
proportionality problem related to our main result is also proposed.
