Monotone increasing function with $f(2a)\leq Cf(a).$

Question

Let $f:\mathbb{R}_{+}\to \mathbb{R}_{+}$ be a mon. increasing function such that $$f(2a)\leq Cf(a).$$ Then it follows that $$a\leq 2b\Rightarrow f(a)\leq Cf(b),$$ but does it also hold that $$a\leq cb\Rightarrow f(a)\leq C'f(b)?$$

Answer 1

We have the obvious result that if $c \leq 2^n$, then for $a \leq cb \leq 2^n b$, $f(a) \leq C^n f(b)$.