Appendix

A holomorphic function m mapping \(\mathbb {C}_{+}\) into \(\mathbb {C}_{+}\) is called a Herglotz–Nevanlinna function  (HN function for short). A Möbius transformation yields a holomorphic function \(\begin{aligned} \widetilde{m}\left( w\right) =m\left( i\dfrac{1+w}{1-w}\right) \end{aligned}\) on \(\Delta =\left\{ \left| w\right| <1\right\} \) .