Suppose $\theta_{1},\dots,\theta_{n-1}$ are semipositive closed $(1,1)$-forms. Then the symmetric bilinear form

$$(f,g)\mapsto\int_{X}f\,dd^{c}g\wedge\theta_{1}\wedge\dots\wedge\theta_{n-1}$$

on $\mathcal{D}(X)$ is negative semidefinite. In particular, for any two model functions $f$, $g$, the following Cauchy-Schwarz inequality holds:

(2.4)

$$\left|\int_{X}f\,dd^{c}g\wedge\theta_{1}\wedge\dots\wedge\theta_{n-1}\right|\leq\\ \left(-\int_{X}f\,dd^{c}f\wedge\theta_{1}\wedge\dots\wedge\theta_{n-1}\right)^{1/2}\,\left(-\int_{X}g\,dd^{c}g\wedge\theta_{1}\wedge\dots\wedge\theta_{n-1}\right)^{1/2}.$$