Pome7FBC12CCC033Lemma 16.15.23.7FBC12CCC033 Let $r\geq 1$ be an integer, let $f_1,\ldots ,f_ r$ be smooth real functions on $X$, let $\delta >0$, and put
Then $F$ is a smooth real function with $\min _ jf_ j-\delta \log r\leq F\leq \min _ jf_ j$, and $D_ hF\leq \max _ j\sup _ XD_ hf_ j$ on $X$.
Proof. Put
The weights $p_ j$ are positive with $\sum _ jp_ j=1$, and $\min _ jf_ j-\delta \log r\leq F\leq \min _ jf_ j$, because the sum lies between $\mathrm e^{-\min _ jf_ j/\delta }$ and $r$ times this number. In local holomorphic coordinates, the chain rule gives $\partial _ iF=\sum _ jp_ j\partial _ if_ j$ and $\partial _{\bar m}p_ j=-\delta ^{-1}p_ j\bigl (\partial _{\bar m}f_ j-\sum _ lp_ l\partial _{\bar m}f_ l\bigr )$; differentiating once more and contracting with $h^{i\bar m}$, we get
The term in parentheses equals $\sum _ jp_ j|\partial f_ j-\sum _ lp_ l\partial f_ l|_ h^2\geq 0$, the variance of the vectors $\partial f_ j$ with respect to the probability weights $p_ j$. Hence $D_ hF\leq \sum _ jp_ jD_ hf_ j\leq \max _ j\sup _ XD_ hf_ j$. $\square$
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