Research Problems in Function Theory — Problem 1.16

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

The spherical derivative is $f^\#(z)=|f'(z)|/(1+|f(z)|^2)$, extended continuously across poles in a local coordinate. Its normalized spherical area is \[ A(r,f)=\frac1\pi\iint_{|z|<r}(f^\#(z))^2\,dx\,dy\quad(z=x+iy). \] In this problem let $n_{<}(r,a,f)$ count points in the open disk $|z|<r$, with multiplicity, and set $n_*(r,f)=\sup_{a\in\widehat{\mathbb C}}n_{<}(r,a,f)$.

Statement recorded in the supplied source.

Find the optimal universal constant $C$ for which every nonconstant meromorphic function in the plane satisfies \[ \liminf_{r\to\infty}\frac{n_*(r,f)}{A(r,f)}\le C. \] In particular, can the known bound $C=e$ be decreased, and can it be replaced by $1$? The source also records the restriction of the $C=1$ question to entire functions. [S2]

Short English statement

Improve the universal bound relating the largest value count to spherical area, and test the bound one, including the entire-function subcase. The source reports a bound below e and a meromorphic counterexample to one; its 2018 update leaves the entire-function case open. [S2]

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2301016 (AMR-022-1016). 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 1.16, its full Update 1.16, and chapter notation; original author TeX. https://arxiv.org/abs/1809.07200.