Let $\mathbb{H}^+=\{z \in \mathbb{C}\mid \Im(z)>0\}$. We say that an analytic $F\colon \mathbb{H}^+\to\overline{\mathbb{H}^+}$ is a Herglotz-Nevanlinna's function.
Question Can it be that $F(z)\in \mathbb{R}$ for some $z \in \mathbb{H}^+$?
I guess that the answer is no, because if this happened then we could find a small loop $\gamma$ around $z$ such that $F\circ \gamma$ slips outside $\overline{\mathbb{H}^+}$, but I'm not sure this is true and how to formalize this little argument.
Thank you.