PomeA9609E0AE7B9Corollary 11.5.3.A9609E0AE7B9 There are constants $0<a\leq b<+\infty $, depending only on $\omega $ and a chosen Gauduchon metric, such that every $u\in \mathrm{PSH}(X,\omega )$ satisfies
Proof. Fix a Gauduchon form $\gamma $ on $X$ and choose positive constants $c,C$ with $c\gamma ^{n-1}\leq \omega ^{n-1}\leq C\gamma ^{n-1}$. Wedging these inequalities with the positive current $\omega +\mathrm{dd}^{\mathrm{c}}u$ and integrating, we obtain, by proposition 11.5.1, the bounds $cA$ and $CA$, where $A=\int _ X\omega \wedge \gamma ^{n-1}>0$. $\square$
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