We can't decuce too much in terms of upper bounds. Note that for any constant $c\ge0$ and any nonnegative $g$ the function $f(x)=cx^{1/2}$ satisfies the inequality.

Note that the inequality is satisfied by the functions $f(x)=cx^{1/2}$, for any $c\ge0$ and any nonnegative $g$. So, in terms of upper bounds, it doesn't really add anything to the information that $f$ is Hölder continuous of exponent 1/2.