Research Problems in Function Theory — Problem 1.34

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, $M(r,f)=\max_{|z|=r}|f(z)|$. Let $n_1(r,a,f)$ be the number of simple zeros of $f-a$ in $|z|\le r$, counted once.

Statement recorded in the supplied source.

Let $f$ have finite order $\rho$. If four distinct finite values $a_1,a_2,a_3,a_4$ satisfy \[ n_1(r,a_j,f)=o(T(r,f))\quad(r\to\infty),\qquad j=1,2,3,4, \] must $\rho$ be an integral multiple of $1/2$ or $1/3$? That is, must \[ \rho\in\{m/2:m\in\mathbb Z,\ m\ge0\}\cup\{m/3:m\in\mathbb Z,\ m\ge0\}? \] This asks whether the known conclusion under bounded simple-zero counts persists under the displayed small-count condition. [S2]

Short English statement

Test whether four values with negligible simple-zero counts impose the classical discrete restriction on the order. The source reports counterexamples, including entire functions of arbitrary positive order. [S2]

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2301034 (AMR-022-1034). 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 1.34, its full Update 1.34, and chapter notation; original author TeX. https://arxiv.org/abs/1809.07200.