Asymptotic behaviour of a concave function

Question

Let $u:\mathbb{R}_+ \to \mathbb{R}$ be a positive, increasing concave function (if needed, I may also assume it is twice differentiable). Are theere any bounds in the order of growth of the function as $x\to +\infty$? I'm looking for an estimate of the form $u(x)=O(...)$.

Answer 1

If you know the derivative at some point $y_0>0$, then for all $y\geq y_0$, $u(y)\leq u(y_0)+u'(y_0)(y-y_0)$ by concavity (you might not be able to do this at $0$, as the derivative may be infinite at $0$, like for $u(x)=\sqrt{x}$). So in general, $u(x)=O(x)$. This is attained for linear functions which are trivially concave.