Dear all,

Suppose we know that $f(x)$ is nonnegative and Hölder continuous at zero with exponents $1/2$. We also know that 

$$
f(x) \le g(x) + \int_0^x \frac{f(y)}{y} d y,\quad\forall x>0,
$$

where $g(x)$ is some nonnegative nice function, for example, $g(x)=\sqrt{x}$. Is it possible to derive a good upper bound for $f(x)$? Apparently, classical [Gronwall's inequality][1] doesn't work since $1/y$ is not integrable around $0$.

Thank you very much for any hints and help! :-)


  [1]: http://en.wikipedia.org/wiki/Gronwall's_inequality