Three Exceptional Values Do Not Force Bounded Type

Nevanlinna’s half-plane problem asks how much control follows when a meromorphic function takes three exceptional values only on the boundary. The expected conclusion was bounded type: if the zero, one, and pole sets are confined to the real line, perhaps the function must have controlled factorization in the upper and lower half-planes. This paper gives a counterexample. It constructs a real meromorphic function $F$ on $\mathbb C$ for which $F^{-1}(\{0,1,\infty\})\subset\mathbb R$, but $F$ is not of bounded type in either half-plane.

Core Contribution

The stronger statement is what makes the construction consequential. For every $a\in\widehat{\mathbb C}\setminus\{0,1,\infty\}$, the paper produces an $a$-point divisor in each half-plane that fails the Blaschke condition. Thus the failure is not tied to one specially chosen value: every nonexceptional target value has enough preimages, distributed in the relevant geometry, to defeat the necessary zero-divisor condition for bounded holomorphic functions.

The result supplies an independent negative answer to a question traced to Nevanlinna’s 1925 work. Postcomposing with a Möbius map transfers the counterexample from the triple $\{0,1,\infty\}$ to any prescribed triple of distinct points on the Riemann sphere. The conclusion is therefore invariant under the natural normalization of three marked values rather than an artifact of that particular choice.

Technical Approach

The proof works through a selected family of points in a covering-theoretic construction. For a fixed nonexceptional value $a$, the authors select points $w_{m,d}=x_{m,d}+iy_{m,d}$ indexed by a level $m$ and a divisor-like index $d\in D_m$. They take $n_m=\lceil4\sqrt{\log m}\rceil$ and obtain, uniformly over $d\in D_m$, the estimates

These points lie in the chosen upper-half-plane domain for sufficiently large $m$. Their distinctness is established by representing the corresponding deck transformations with matrices in $\mathrm{PSL}_2(\mathbb Z)$: equality of selected points forces equality of the transformation data and hence of the pair $(m,d)$. At level $m$, the construction yields exactly $\varphi(2m)$ selected points, where $\varphi$ is Euler’s totient function.

Local conformality of the maps involved turns the selected $w_{m,d}$ into distinct simple $a$-points $z_{m,d}$ of $F$ in the upper half-plane. The analytic bridge is a lower bound for the upper-half-plane Green function. The paper derives

for all sufficiently large levels. This is the formal quantity that tests whether the divisor can satisfy the Blaschke condition.

Results and Analysis

The divergence argument is arithmetic rather than asymptotic hand-waving. Using the classical estimate

the authors show that the dyadic blocks contribute enough totient-weighted mass to make

Combining this with the Green-function bound gives a divergent Green sum over the selected $a$-points. Since their real parts tend to infinity while the Green-function base point remains fixed, this selected subset already violates the Blaschke condition in $\mathbb H^+$. Reflection and the real symmetry $F(\overline z)=\overline{F(z)}$ give the corresponding lower-half-plane statement.

The final bounded-type contradiction is direct. If $F=U/V$ in the upper half-plane with bounded holomorphic $U$ and $V$, then for any $a\notin\{0,1,\infty\}$ the bounded holomorphic function $U-aV$ has the selected $a$-points as zeros. Their divergent Green sum is incompatible with the Blaschke condition for zeros of a bounded holomorphic function. The proof therefore supports the central claim at the level appropriate for a pure complex-analysis result: an explicit construction plus a chain of geometric, arithmetic, and function-theoretic estimates. Its scope is narrow by design—it settles a structural implication, not a numerical or applied benchmark—but within that scope the argument addresses both half-planes and every nonexceptional value.