PomeProposition 14.19.36.92CEFF8A020B Let $u,v\in C^2(X,\mathbb R)$, and set $A=\omega +\mathrm{dd}^{\mathrm{c}}u$ and $B=\omega +\mathrm{dd}^{\mathrm{c}}v$. If $A,B\geq 0$ and $A^ n=cB^ n$ for a constant $c>0$, then $c=1$.
Proposition 14.19.3692CEFF8A020B
Proof. Fix $\varepsilon >0$. At a maximum point of $u-(1+\varepsilon )v$, the complex Hessian of this function is nonpositive, so that
\[ A+\varepsilon \omega \leq (1+\varepsilon )B. \]
At this point $B>0$, because $A+\varepsilon \omega >0$. Monotonicity of the determinant and $A^ n=cB^ n$ give $c\leq (1+\varepsilon )^ n$. Letting $\varepsilon \downarrow 0$ gives $c\leq 1$, and interchanging the two potentials gives $c^{-1}\leq 1$. $\square$
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