Research Problems in Function Theory — Problem 1.8

Definitions and mathematical statement

For a nonconstant meromorphic function $f$, let $n(t,a)$ count solutions of $f(z)=a$ in $|z|\le t$, with multiplicity; $a=\infty$ means poles. Put \[ N(r,a)=\int_0^r\frac{n(t,a)-n(0,a)}t\,dt+n(0,a)\log r, \quad T(r,f)=N(r,\infty)+\frac1{2\pi}\int_0^{2\pi}\log^+|f(re^{i\theta})|\,d\theta, \] where $\log^+x=\max(0,\log x)$. At a boundary radius $R\in\{1,\infty\}$ with $T(r,f)\to\infty$, define the deficiency $\delta(a,f)=1-\limsup_{r\to R}N(r,a)/T(r,f)$. A deficient value has positive deficiency. The order and lower order in the plane are respectively $\rho=\limsup_{r\to\infty}\log T(r,f)/\log r$ and $\lambda=\liminf_{r\to\infty}\log T(r,f)/\log r$. Take finite nonintegral $\rho>1$, put $q=\lfloor\rho\rfloor$, and define \[ K(\rho)=\begin{cases} |\sin\pi\rho|/(q+|\sin\pi\rho|),&q<\rho<q+\tfrac12,\\ |\sin\pi\rho|/(q+1),&q+\tfrac12\le\rho<q+1. \end{cases} \] For every pair of distinct values $a,b\in\widehat{\mathbb C}$, is \[ \limsup_{r\to\infty}\frac{N(r,a)+N(r,b)}{T(r,f)}\ge K(\rho)? \] The associated rigidity question asks whether equality forces $\lambda=\rho$, namely equality of lower order and order. This meaning of regular growth does not assert an asymptotic formula $T(r,f)\sim cr^\rho$. Both counting functions are evaluated at the same radii before taking the upper limit.

Short English statement

Must every pair of distinct values of a meromorphic function occur with the proposed sharp frequency, and does equality force equality of order and lower order?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2301008 (AMR-022-1008), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 1.8. Original source indexed by AMR-022-1008. https://arxiv.org/abs/1809.07200.