Research Problems in Function Theory — Problem 1.27

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a nonconstant meromorphic function $f$ on $\mathbb C$, let $n(r,a,f)$ count the solutions of $f(z)=a$ in $|z|\le r$, with multiplicity; $a=\infty$ counts poles. Let $\bar n$ count each point once. The integrated count is \[ N(r,a,f)=n(0,a,f)\log r+\sum_{0<|z|\le r,\ f(z)=a}\log(r/|z|), \] where the sum repeats each point according to multiplicity; $\bar N$ has the same formula without repetitions. Write $\log^+x=\max(\log x,0)$ and \[ m(r,\infty,f)=\frac1{2\pi}\int_0^{2\pi}\log^+|f(re^{it})|\,dt,\quad m(r,a,f)=\frac1{2\pi}\int_0^{2\pi}\log^+\frac1{|f(re^{it})-a|}\,dt\quad(a\in\mathbb C). \] The Nevanlinna characteristic is $T(r,f)=m(r,\infty,f)+N(r,\infty,f)$. Its order and lower order are \[ \rho(f)=\limsup_{r\to\infty}\frac{\log T(r,f)}{\log r},\qquad \lambda(f)=\liminf_{r\to\infty}\frac{\log T(r,f)}{\log r}. \] The deficiency is $\delta(a,f)=1-\limsup_{r\to\infty}N(r,a,f)/T(r,f)$, and the reduced-count index is $\Theta(a,f)=1-\limsup_{r\to\infty}\bar N(r,a,f)/T(r,f)$. A value is deficient if $\delta(a,f)>0$. An entire function is holomorphic on all of $\mathbb C$; for an entire function, $M(r,f)=\max_{|z|=r}|f(z)|$.

For a compact set $K\subset\mathbb C$, its logarithmic capacity is \[ \operatorname{cap}(K)=\exp\!\left(-\inf_\mu\iint\log\frac1{|z-w|}\,d\mu(z)\,d\mu(w)\right), \] where $\mu$ ranges over Borel probability measures supported on $K$; $\exp(-\infty)=0$ and the empty set has capacity zero. A set has inner capacity zero if every compact subset has logarithmic capacity zero. A set $E\subset[1,\infty)$ has finite logarithmic measure when $\int_E dr/r<\infty$.

Statement recorded in the supplied source.

Let \[ E_f=\{a\in\mathbb C:\lim_{r\to\infty}m(r,a,f)=\infty\}. \] Determine the possible size of $E_f$, and the restrictions on it, in each of the following classes:

1. Entire functions of order $1/2$ and mean type, meaning $0<\limsup_{r\to\infty}\log M(r,f)/\sqrt r<\infty$.
2. Meromorphic functions of order $\rho$ with $0\le\rho\le1/2$.

In particular, the question concerns capacity as well as cardinality; the source recalls that such proximity-divergence sets have capacity zero. [S2]

Short English statement

Find the possible proximity-divergence sets in the two specified small-order classes. The update supplies meromorphic constructions covering any prescribed set of capacity zero, without claiming that the prescribed set is necessarily the exact divergence set. [S2]

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2301027 (AMR-022-1027). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Eleanor F. Lingham, Research Problems in Function Theory, Fiftieth Anniversary Edition (2018), arXiv:1809.07200. Problem 1.27, its full Update 1.27, and chapter notation; original author TeX. https://arxiv.org/abs/1809.07200.