When a meromorphic function that omits three values has a bounded type

Suppose that a function $F$ is meromorphic in the domain $\mathbb{H}(-m)=\{z:\mathrm{I}\mathrm{m}z>-m(\mathrm{R}\mathrm{e}z)\}$, where $m$ is an even, positive, and continuous function that does not increase on ${\mathbb{R}}_{\ge 0}$, 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 the function $m$ is convergent. On the other hand, if the logarithmic integral of $m$ diverges, there exists a function $F$ meromorphic in $\mathbb{H}(-m)$, that omits there three distinct values, and which is of unbounded type in the upper half-plane. This result is motivated by a century old question originating with Rolf Nevanlinna, which remains unsettled.