# Stieltjes transform of a compactly supported measure : behaviour at the boundary

I ask here the same question I asked on Mathematics to maybe reach other poeple :

I am studying the Stieltjes transform $$G_\mu(z) = \int_a^b \frac{1}{z-s} d \mu(s)$$ of some positive finite measure $$\mu$$ which has the compact support $$[a,b]$$. We also assume that $$\mu$$ is absolutely continuous with respect to the Lebesgue measure on $$\mathbb{R}$$. Also we may assume that its density is Hölder continuous on $$[a,b]$$.

I would like to show that $$G_\mu(z) = O(log z),$$ when $$z \rightarrow a$$ and $$z \rightarrow b$$ (e.g. for $$a$$, I want to show that there exists $$C,\delta > 0$$ such that $$|z-a| \leq \delta \implies |G_\mu(z)| \leq C |log z|$$). I intuitively feel like it is true since it behaves as the primitive of $$1/z$$ but I can't prove it rigourously. Could you provide me some hints ?