Let H=z∈C:operatornameImz>0, and let N(H) be the class of meromorphic functions on H that are quotients of two bounded analytic functions on H. Is it true that every nonconstant meromorphic function F on C satisfying F^-1(0,1,∞)⊆R also satisfies F mathbin|_H∈ N(H)?

Nevanlinna’s half-plane problem

Let $\mathbb{H}=\{z\in\mathbb{C}:\operatorname{Im}z>0\}$ , and let $N(\mathbb{H})$ be the class of meromorphic functions on $\mathbb{H}$ that are quotients of two bounded analytic functions on $\mathbb{H}$ . Is it true that every nonconstant meromorphic function $F$ on $\mathbb{C}$ satisfying $F^{-1}(\{0,1,\infty\})\subseteq\mathbb{R}$ also satisfies $F\mathbin{|}_{\mathbb{H}}\in N(\mathbb{H})$ ?

References

Primary source

arXiv

Additional references

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

Posted 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A new preprint claims a counterexample that overturns this century-old problem, but the result has not yet been independently confirmed.

Nevanlinna’s problem asks whether omitting three values in a half-plane forces a meromorphic function to belong to the Nevanlinna class there. A new preprint claims the implication fails by constructing a nonconstant meromorphic function on the entire plane.

August 2026 counterexample

A new preprint, A counterexample to Nevanlinna's century-old half-plane problem, claims an entire-plane meromorphic counterexample, which would settle the problem negatively. An earlier 2026 preprint proved sharp-looking results for variable half-planes but explicitly said the original question remained unsettled; the newer counterexample claim supersedes that as the latest reported development.

Current status (as of August 2026): A preprint claims the problem is solved negatively by a three-value-omitting counterexample, but independent verification is not recorded; absent confirmation, the result remains unverified.

Sources

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