Misha Sodin

When a meromorphic function that omits three values is of bounded type

Suppose that a function F is meromorphic in the domain H(−m)={z: Imz>−m(Rez)},
where m is an even, positive, and continuous function that does not increase
on the positive ray, and suppose that F omits there three distinct values. Then F is of
bounded type in the upper half-plane (i.e., is represented there as a quotient
of two bounded analytic functions), provided that the logarithmic integral of m
converges.

On the other hand, if the logarithmic integral of m diverges, there exists a function
F meromorphic in H(−m), that omits there three distinct values, and which is of
unbounded type in the upper half-plane. The existence part uses the elliptic modular
function, a little number theory, and a bit of harmonic measure.

This result is motivated by a century-old question originating with Rolf Nevanlinna.

Joint work with Alexandre Eremenko and Aleksei Kulikov (https://arxiv.org/abs/2604.06136)