Research Problems in Function Theory — Problem 1.28

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, $M(r,f)=\max_{|z|=r}|f(z)|$.

A path to infinity is a continuous curve $\gamma:[0,\infty)\to\mathbb C$ with $|\gamma(t)|\to\infty$. A value $a\in\widehat{\mathbb C}=\mathbb C\cup\{\infty\}$ is an asymptotic value of $f$ if $f(\gamma(t))\to a$ along such a path, using the spherical topology when $a=\infty$.

Statement recorded in the supplied source.

For finite-order meromorphic functions $f$, determine universal upper restrictions, if any, on \[ \mathcal A(f)=\{a\in\widehat{\mathbb C}:a\text{ is an asymptotic value of }f\}. \] In particular, can this set be all of $\widehat{\mathbb C}$ even when $\rho(f)<\infty$? The question concerns meromorphic functions and all asymptotic values, rather than only direct singularities of an inverse function. [S2]

Short English statement

Ask how large the asymptotic-value set of a finite-order meromorphic function can be. The source reports examples of positive and zero order with the entire Riemann sphere as that set. [S2]

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2301028 (AMR-022-1028). 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 1.28, its full Update 1.28, and chapter notation; original author TeX. https://arxiv.org/abs/1809.07200.