Lemma 7.9.1.0FF4AFFE0ED0 Let $u,v$ be locally bounded plurisubharmonic functions on $\Omega $, and let $F,G\ge 0$ be continuous functions on $\Omega $ with $(\mathrm{dd}^{\mathrm{c}}u)^ n\ge F^ n\eta ^ n$ and $(\mathrm{dd}^{\mathrm{c}}v)^ n\ge G^ n\eta ^ n$. Then, for $0\le t\le 1$ and $c\ge 0$,
\[ \bigl (\mathrm{dd}^{\mathrm{c}}((1-t)u+tv+c|z|^2)\bigr )^ n \ge \bigl ((1-t)F+tG+c\bigr )^ n\eta ^ n. \]
Moreover, $(\mathrm{dd}^{\mathrm{c}}\max \{ u,v\} )^ n\ge \min \{ F,G\} ^ n\eta ^ n$.
Proof.
Let $B$ be a positive definite Hermitian matrix with $\det B=1$. By the implication from (1) to (3) in proposition 7.8.3, $L_ Bu\ge nF$ and $L_ Bv\ge nG$ in the sense of distributions. Moreover $L_ B|z|^2=\operatorname {tr}B\ge n(\det B)^{1/n}=n$ by the arithmetic–geometric mean inequality for the eigenvalues of $B$. Hence
\[ L_ B\bigl ((1-t)u+tv+c|z|^2\bigr ) \ge n\bigl ((1-t)F+tG\bigr )+c\operatorname {tr}B \ge n\bigl ((1-t)F+tG+c\bigr ), \]
and the implication from (3) to (1), applied with the continuous density $((1-t)F+tG+c)^ n$, gives the first assertion.
For the second, let $q$ be an upper test function for $\max \{ u,v\} $ at a point $x$. If $u(x)\ge v(x)$, then $q(x)=u(x)$ and $q\ge \max \{ u,v\} \ge u$ near $x$, so $q$ is an upper test function for $u$ at $x$, and the implication from (1) to (2) gives $\det \mathcal H(q)(x)\ge F(x)^ n\ge \min \{ F,G\} (x)^ n$. The case $v(x)\ge u(x)$ is symmetric. The implication from (2) to (1) now gives the assertion.
$\square$
Pome
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