Research Problems in Function Theory — Problem 2.7

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 $a=\infty$.For a locally rectifiable path $\gamma$ starting at $z_0$, put $\tau_r=\inf\{t\ge0:|\gamma(t)|=r\}$ for $r>|z_0|$, and let $\ell_\gamma(r)$ be the arc length from $\gamma(0)$ to $\gamma(\tau_r)$; length may be infinite. Also let $L_\gamma(r)$ be the total arc length of portions of the path inside $|z|\le r$, counted along its parametrization.

Statement recorded in the supplied source.

For nonconstant entire functions of finite order $\rho$, determine the growth bounds that can be guaranteed for the length of some asymptotic path on which $f\to\infty$, including the length of its portions in $|z|\le r$. The update makes the first-intersection measure $\ell_\gamma(r)$ explicit. In particular, can one always achieve $\ell_\gamma(r)=O(r)$, and what sharper dependence on $\rho$ is possible beyond the known existence, for each $\varepsilon>0$, of a path with \[ \ell_\gamma(r)=O(r^{1+\rho/2+\varepsilon})? \] The original also asks about lengths inside the disk, measured by $L_\gamma(r)$; first-intersection bounds alone need not control later re-entry. [S2]

Short English statement

Bound how short a path to the asymptotic value infinity can be for an entire function of finite order, distinguishing first-intersection length from total length in a disk. The source reports that linear length can fail and gives the displayed polynomial bound. [S2]

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302007 (AMR-022-2007). 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 2.7, its full Update 2.7, and chapter notation; original author TeX. https://arxiv.org/abs/1809.07200.