The Monotone Density Theorem for regularly varying functions says, in essence:

**Theorem** (Monotone Density Theorem). *Let $f$ be a differentiable regularly varying real-function of index $\rho$ well-defined in $[a,+\infty)$ for some $a\geqslant 0$. If $f'$ is asymptotically monotone, then $f'$ is of regular variation of index $\rho-1$.*

A proof (with a slightly different statement) may be found in p. 39 of Bingham's book *Regular Variation*. Although, this proof makes strong use of Karamata's characterization theorem, and offers no clue how to extend it to $O$-regularly varying functions.

A positive real-function $f$ defined in $[a,+\infty)$ is said to be of *regular variation* if for all $\lambda > 0$ holds
$$f(\lambda x) \sim g(\lambda)f(x)~\text{ as }~x\to +\infty,~\text{with }~ g(\lambda) \in (0,+\infty).$$

An *$O$-regular variation* is the weaker condition that for all $\lambda > 0$ holds
$$f(\lambda x) \asymp f(x)~\text{ as }~x\to +\infty.$$

The "O-version of Monotone Density Theorem" described in p. 119 of Bingham's book (Proposition 2.10.3) seems to be a far generalization of this fact, but I had a little trouble even to understand how to effectively handle the definition of Matuszewska indices (although *$O$-regularity* clearly implies $BI\cap PI$). The propostion is as follows:

**Proposition 2.10.3.** [Bingham's book] *Let $f$ be a differentiable real-function well-defined in $[a,+\infty)$ for some $a\geqslant 0$. Assume $f'$ is asymptotically positive and $f'\in BI\cup BD$. If $f \in BI\cap PI$, then $f'(x)\asymp f(x)/x$ as $x\to+\infty$.*

The acronyms stands for $BI(BD)$ = bounded increase (decrease), $PI(PD)$ = positive increase (decrease) and are defined in terms of Matuszewska indices (p.68) in p.71 of Bingham's book.

So my question is:

Does Proposition 2.10.3 of Bingham implies the Monotone Density Theorem assuming the same hypotheses but changing

regularto$O$-regular? If so, is there an easier way to do this?