Solution

Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 12 / 4 / a / Solution

{title2=$|\operatorname{Log}(1-w)|\le2|w|\quad (|w|\le1/2)$}

The <power series> $\operatorname{Log}(1-w)=-\sum_{j\ge1}w^j/j$ proves this estimate. There is no global bound $|\operatorname{Log}(1-w)|\le C|w|$, since the logarithm diverges as $w\to1$. The different estimate $\log|1-w|\le\log(1+|w|)\le|w|$ is valid wherever $w\ne1$, and is sufficient for growth bounds on <canonical products>.

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 Incoming links: Bounds for the principal logarithm near one