an analytic function on the upper-half plane with values in the lower half-plane. Cauchy transform determines uniquely the measure and there is an inversion formula called Stieltjes inversion formula, namely for $h_{\varepsilon}(t)=-\tfrac{1}{\pi}{\text{\normalfont Im}}\, G_\mu(t+i{\varepsilon})$ one has $${{\rm d}}\mu(t)=\lim_{{\varepsilon}\to 0^+} h_{\varepsilon}(t)\,{{\rm d}}t,$$ where the limit is taken in the weak topology.

2.  For a compactly supported measure $\mu$ one can define in a neighbourhood of the origin so called $R$-transform by $$R_\mu(z)=G_\mu^{\langle -1 \rangle}(z)-\frac{1}{z},$$ where by $G_\mu^{\langle -1 \rangle}$ we denote the inverse under composition of the Cauchy transform of $\mu$.\
    The relevance of the $R$-transform for free probability comes form the fact that it linearizes free convolution, that is $R_{\mu\boxplus\nu}=R_\mu+R_\nu$ in a neighbourhood of zero.

The $R$-transform of fGIG distribution is given by $$\label{eq:R_F}
\begin{split}
r_{\alpha,\beta,\lambda}(z) 
&= \frac{-\alpha + (\lambda+1)z + \sqrt{f_{\alpha,\beta,\lambda}(z)}}{2z(\alpha-z)} 
\end{split}$$ in a neighbourhood of $0$, where the square root is the principal value, $$\label{eq:pol_fpar} 
f_{\alpha,\beta,\lambda}(z)=(\alpha+(\lambda-1)z)^2-4\beta z (z-\alpha)(z-\gamma),$$ and $$\begin{aligned}
\gamma=\frac{\alpha^2 a b+\frac{\beta^2}{ab}-2\alpha\beta\left(\frac{a+b}{\sqrt{ab}}-1\right)-(\lambda-1)^2}{4\beta}.\end{aligned}$$ Note that $z=0$ is a removable singular point of $r_{\alpha,\beta,\lambda}$. Observe that in terms of $A,B$ defined by we have $$\begin{aligned}
\gamma &= 2\frac{\lambda A^2 + A B - 2B^2}{B(B-A)^2}. \end{aligned}$$ It is straightforward to observe that implies $A (\lambda A+B)<2 A B<2B^2$, thus we 