A continuous function $L :\mathbb{R}_+ \to \mathbb{R}_+$ is called regularly varying (at infinity) if for all $a > 0$,

$$ \lim_{x\to \infty} \frac{L(ax)}{L(x)}= g(a) $$

for some function $g(a)>0$. If $g(a)=1$ then $L$ is said to be slowly varying. This definition is due to Karamata, and a very good reference for this class of functions is the book:

Bingham, N. H.; Goldie, C. M.; Teugels, J. L., Regular variation., Encyclopedia of Mathematics and its Applications, 27. Cambridge etc.: Cambridge University Press

I recently had to deal with a class of functions $S :\mathbb{R}_+ \to \mathbb{R}_+$ such that, for all $a>0$

$$ \lim_{x\to \infty} \frac{S(x^a)}{S(x)}= g(a), \tag{$\star$} $$

for some $g(a)>0$. It is clear that a function satisfyes $(\star)$ if and only if $S(x) = L(\log(x))$ for some regularly varying function $L$. Furthermore, one can show that $S$ is slowly varying.

Has the class of functions verying $(\star)$ been studied in the literature? Can someone point to a reference? If not, if you were to give a name to this class of functions, what would you suggest?

  • $\begingroup$ How about "logarithmically regularly/slowly varying"? $\endgroup$ – Greg Martin Nov 20 '18 at 21:58
  • $\begingroup$ I do not favour this name. For example, a function $f$ is log-concave if $\log f$ is concave. So A log-slowly varying function would be a function such that $\log f$ is slowly varying. Here it is quite the opposite. But I will add this to the list of names anyway. $\endgroup$ – Raziel Nov 21 '18 at 8:30

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