Given strictly concave function $f(t)$ that satisfies $f'(t)>0$, $f'(t)=o(1)$ (i.e. $\lim\limits_{t\to\infty}f'(t)=0$) , and $f'(t)=\omega\left(t^{-1}\right)$ (i.e. $\lim\limits_{t\to\infty}tf'(t)=\infty$) , we denote $F=\exp\left(f\left(t\right)\right)$. Let $H(t)$ be a solution of $$ \dot{H}=F $$ Can we approximate H by F?

Specifically, I want to show that $$ \lim_{s\to\infty}\frac{H^{-1}(s)}{F^{-1}(s)}=1\,, \ \ \ \ (a) $$ where $F^{-1}$ and $H^{-1}$ are the inverse functions of $F$ and $H$, respectively.

If necessary I can add additional assumptions on $f(t)$, though I would like to keep it as general as possible.

* In this question, $(a)$ was proved for $f'(t)=\Omega(1)$ (without the assumption that $f(t)$ is strictly concave). I'm interested in the case that $f'(t)=o(1)$.