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Proposition 14.7.1C52E44D9C563

Proposition 14.7.1.C52E44D9C563 Let $\lambda >0$, let $F$ be smooth, and let $v,w$ be smooth strictly $\omega $-plurisubharmonic functions. If

\[ \omega _ v^ n\geq \mathrm{e}^{\lambda v+F}\omega ^ n, \qquad \omega _ w^ n\leq \mathrm{e}^{\lambda w+F}\omega ^ n, \]

then $v\leq w$. In particular, two smooth solutions coincide.

Proof. Suppose that $v-w$ has a positive maximum at a point $x$. Then $\omega _ v(x)\leq \omega _ w(x)$, and taking determinants and applying the two assumptions gives $\mathrm{e}^{\lambda v(x)+F(x)}\leq \mathrm{e}^{\lambda w(x)+F(x)}$, which contradicts $v(x)>w(x)$. For two solutions, apply the comparison in both orders. $\square$

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