Concavity follows from (6.4) and Proposition 2.21. Monotonicity is a consequence of (6.3), and the last equation follows from (6.2) since $(\omega+dd^{c}\varphi)^{j}\wedge\omega^{n-j}$ is a probability measure for each $j$ thanks to Proposition 2.19 and the normalization $\{\omega\}^{n}=1$. ∎