Proposition 14.2.2853DC10A2ADA

Proof. At a maximum point $x_+$ of $u$, the real $(1,1)$-form $\mathrm{dd}^{\mathrm{c}}u$ is nonpositive, so $0<\omega _ u\leq \omega $ there. Hence $\omega _ u^ n(x_+)\leq \omega ^ n(x_+)$, and $b\leq \log (\omega ^ n/\mathrm{d}V)(x_+)-F(x_+)$. At a minimum point the reverse argument gives the lower bound. $\square$