I am trying to find smooth functions $f : \mathbb{R}_+ \to \mathbb{R}_+$ such that the quantity $$\Delta_f(x) := 2f(x)-f(2x)$$ is positive for $x$ large enough and has the greatest asymptotic growth.

It seems clear that one cannot go beyond a linear growth, since $\Delta_f$ vanishes for linear functions and will likely be negative for super-linear ones. However, one can construct many examples of almost linear asymptotic growths.

For example, plugging $f(x) := x^a$ for some $a\in(0,1)$ yields $\Delta_f(x) = (2-2^a) x^a$.

Choosing $f(x) := \frac{x}{\ln x}$ yields $\Delta_f(x) = 2 \ln 2 \frac{x}{(\ln x)^2}$ which has a larger asymptotic growth.

What would be your candidates for even larger asymptotic growths? And is there a way to prove a (sublinear) a priori upper bound on $\Delta_f$?