If $u''>0$ in $\mathbf{R}^+$ then $u$ is unbounded? If $u$ is a positive function such that $u''>0$ in the whole $\mathbf{R}^+$ then $u$ is unbounded?
In fact, I know that if $u''>0$ then $u$ is strictly convex. I think that implies $u$ is coercive. I want to prove it.

Consider the function
$$f(x):={1\over 1+x}\qquad(x>0)\ .$$
Then $0<f(x)<1$ and $f''(x)>0$ for all $x>0$.
