Corollary 7.12.87071C55A0E4E

Proof. By additivity and symmetry of the products, $(\mathrm{dd}^{\mathrm{c}}(u+v))^ n=\sum _{k=0}^ n\binom nk(\mathrm{dd}^{\mathrm{c}}u)^ k\wedge (\mathrm{dd}^{\mathrm{c}}v)^{n-k}$. Bound the terms with $1\leq k\leq n-1$ by corollary 7.12.6 and the terms with $k=0,n$ by the hypotheses; the binomial formula then gives the assertion. The density on the right is locally $\mu $-integrable, since $f^{k/n}g^{(n-k)/n}\leq f+g$ by the weighted arithmetic–geometric mean inequality, and $f$ and $g$ are locally $\mu $-integrable because $f\mu \leq (\mathrm{dd}^{\mathrm{c}}u)^ n$ and $g\mu \leq (\mathrm{dd}^{\mathrm{c}}v)^ n$. $\square$