Research Problems in Function Theory — Problem 2.5

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 angles $\theta$ modulo $2\pi$ and $0<\varepsilon<\pi$, put \[ D(\theta,\varepsilon)=\{z\ne0:\min_{j\in\mathbb Z}|\arg z-\theta+2\pi j|<\varepsilon\}. \] Saying that $f$ assumes $a$ infinitely often in this sector means that the equation $f(z)=a$ has infinitely many distinct solutions there, with poles interpreted as $\infty$-points.

Statement recorded in the supplied source.

For an entire function $f$, define \[ E_f=\{a\in\mathbb C:\text{for every }\theta\text{ and every }\varepsilon>0, f(z)=a\text{ has infinitely many solutions in }D(\theta,\varepsilon)\}. \] Determine the possible sets $E_f$. In particular, is there an entire $f$ with $|E_f|=2$? The question is about occurrence in every sector, rather than only occurrence somewhere in the plane. [S2]

Short English statement

Describe the values an entire function takes infinitely often in every angular sector, and ask whether there can be exactly two. The source reports a construction with A contained in E contained in the closure of A for every countable A; for a two-point A this answers the particular question affirmatively. [S2]

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302005 (AMR-022-2005). 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 2.5, its full Update 2.5, and chapter notation; original author TeX. https://arxiv.org/abs/1809.07200.