A counterexample to Nevanlinna's century-old half-plane problem

Abstract

Let $\mathbb{H}=\{z\in\mathbb{C}:\operatorname{Im}z>0\}$, and let $N(\mathbb{H})$ denote the Nevanlinna class in $\mathbb{H}$, consisting of meromorphic functions representable as quotients of two bounded analytic functions in $\mathbb{H}$. We construct a nonconstant meromorphic function $F$ on $\mathbb{C}$ such that $F^{-1}(\{0,1,\infty\})\subset\mathbb{R}$ and $F|_{\mathbb{H}}\notin N(\mathbb{H})$. Thus, omitting three distinct values of the Riemann sphere in a half-plane does not force a meromorphic function on $\mathbb{C}$ to be of bounded type there. This provides a counterexample to Nevanlinna's century-old half-plane problem.