Research Problems in Function Theory — Problem 2.8

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.

Let $f$ be meromorphic in the plane and have lower order $\lambda(f)>1/2$. Suppose the number $n_\infty^{<}(r,f)$ of poles in $|z|<r$, counted with multiplicity, satisfies \[ n_\infty^{<}(r,f)=O(r^k)\quad\text{for some real }k<1/2. \] Must there be a single path $\gamma$ to infinity such that, for every integer $m\ge0$, \[ \frac{|f(\gamma(t))|}{|\gamma(t)|^m}\longrightarrow\infty\quad(t\to\infty)? \] The same path must work for every $m$. This expands equation (2.1) cross-referenced in the source; the growth hypothesis is lower order, not order. [S2]

Short English statement

With fewer than square-root-many poles and lower order above one-half, must one path support growth faster than every power? The source update discusses sufficient conditions for the weaker conclusion that infinity is an asymptotic value. [S2]

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302008 (AMR-022-2008). 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 2.8, its full Update 2.8, and chapter notation; original author TeX. https://arxiv.org/abs/1809.07200.