I am not sure if this question is too specific on notations (I think the question is intuitive, but basically the only reference I know with this kind of notations is Bingham, Goldie & Teugels book *Regular Variation*), so let me introduce some things.

Firstly, let $f$ be a measurable positive real function defined in $[a, +\infty)$ for some real $a \geqslant 0$.

**- MATUSZEWSKA INDICES:**

Say that $f$ is

almost increasingwhen there is $M\geqslant a$ and $m>0$ such that $$ f(y) \geqslant mf(x),\quad \forall y\geqslant x\geqslant M. $$The definition of

almost decreasingis similar. Now define theupper($\alpha$) andlower($\beta$)Matuszewska indicesof $f$ as$$ \alpha(f) := \inf\{\alpha \in \mathbb{R} : x^{-\alpha} f(x) \text{ is almost decreasing}\},$$ $$ \beta(f) := \sup\{\beta \in \mathbb{R} : x^{-\beta} f(x) \text{ is almost increasing}\}.$$

These indices basically bound $f$ between two growth orders "neglecting" some slowly varying functions, i.e. $$ \frac{x^{\beta(f)}}{L_1(x)} \ll f(x) \ll x^{\alpha(f)} L_2(x),$$ for some non-decreasing $L_1(x), L_2(x) = o(x^{\varepsilon})$ for all $\varepsilon > 0$.

**- $O$-REGULAR VARIATION:**

$f$ is said to be

$O$-regularly varyingwhen $$ \forall \lambda > 0, f(\lambda x) = \Theta(f(x)),$$ that is, for all $\lambda >0$ there is $M=M(\lambda)>0$ such that: $$ \forall x > M, \exists c_1=c_1(\lambda),c_2 = c_2(\lambda)~: ~c_1 f(x) \leqslant f(\lambda x) \leqslant c_2f(x).$$This is (un?)surprisingly equivalent to say that $f$ has both Matuszewska indices finite (in $(-\infty, +\infty)$).

My question is:

Q:Let $f$ be a measurable, positive,increasingreal function in $[0,\infty)$. If $f$ is $O$-regularly varying and $f(x) \gg x^{1/n}$ for some integer $n \geqslant 1$, then the lower Matuszewska index $\beta(f)$ is $> 1/2n$?

To me it is a somewhat intuitive statement (at a first glance I would even say that $\beta(f) \geqslant 1/n$), but I could not prove it or find it explicitly stated on Bingham's et al. book and other references (Feller's Probability books and some papers). I was looking for some help to develop a better intuition on this statement (being it true or false!). Any tips and/or references would be welcome.

Thanks!

**Disclaimer:** What I tried: https://i.sstatic.net/qGooL.png (These calculations didn't help...)

(Here $A\equiv f$ and $h\equiv n$. Sorry, too lazy to fix!)