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