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

Question

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 ?

Answer 1

A detailed study of Cauchy integrals and a proof of the result are in

Complex Variables, M.J. Ablowitz, A.T. Fokas, Cambridge University Press,

Chapter 7 Riemann-Hilbert problems, Section 7.2 p.518 Cauchy Type Integrals