ca.classical analysis and odes – Increasing concave functions bounded between linear and quadratic

Are there any explicit examples of twice-differentiable functions $g$, defined on the nonnegative real axis $(0,infty)$, such that

  • $g'(x) ge0$
  • $g”(x)le0$,

and such that

  • For $xge K$ for any fixed number $Kge1$ (you have the freedom to choose any $Kge1$ here), we have the property $|g”(x)| > C/x$, preferably $|g”(x)| ge C/x^{1-epsilon}$, for some $epsilon>0$?

Here $C$ is a constant that can be any nonzero number.