The (seemingly) universal example of a slowly varying function in extreme value theory is a power of the natural log, e.g. $\lambda (\ln (x))^{a}$ for $x > 0$, $\lambda \ge 0$, and $a \ge 0$. In view of Karamata's representation theorem, can we say all slowly varying functions $f : [0,\infty) \rightarrow \mathbb{R}$ satisfy $|f(x)|$ $\le$ $\lambda (\ln (x))^{a}$ as $x$ $\rightarrow$ $\infty$ for $\lambda \ge 0$ and $a \ge 0$?
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$\begingroup$The answer is no.
There are slowly varying functions that grow faster than any power of $\log x$ but more slowly than any power of $x$. Examples are $f(x) = \exp((\log x)^b)$ with $0 < b < 1$.
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