I'm looking for a continuous, strictly increasing, strictly convex function $f: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$, with $f(0)=0$, and such that
$ \lim_{x \rightarrow\infty} \frac{f(x)}{x} \leq c $
for some $c \in \mathbb{R}_\geq 0$.
Suggestions?