Research Problems in Function Theory — Problem 1.24

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 \[ A(r,f)=\frac1\pi\iint_{|z|<r}(f^\#(z))^2\,dx\,dy\quad(z=x+iy). \]

For a compact set $K\subset\mathbb C$, its logarithmic capacity is \[ \operatorname{cap}(K)=\exp\!\left(-\inf_\mu\iint\log\frac1{|z-w|}\,d\mu(z)\,d\mu(w)\right), \] where $\mu$ ranges over Borel probability measures supported on $K$; $\exp(-\infty)=0$ and the empty set has capacity zero. A set has inner capacity zero if every compact subset has logarithmic capacity zero. A set $E\subset[1,\infty)$ has finite logarithmic measure when $\int_E dr/r<\infty$.

Statement recorded in the supplied source.

For every nonconstant meromorphic function $f$ in the plane, can its nonintegrated value counts be asymptotically compared with their spherical average using one exceptional set of radii independent of the value? In the precise form recorded in the update, do there exist $E\subset[1,\infty)$ of finite logarithmic measure and $Z\subset\widehat{\mathbb C}$ whose finite part has inner capacity zero such that \[ \lim_{\substack{r\to\infty\\r\notin E}}\frac{n(r,a,f)}{A(r,f)}=1 \quad\text{for every }a\in\widehat{\mathbb C}\setminus Z? \] The same $E$ must work for all nonexceptional values. [S2]

Short English statement

Compare ordinary value counts with their mean spherical area, allowing a small value set and a common small set of radii. The source credits Miles with the stated affirmative form. [S2]

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2301024 (AMR-022-1024). 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 1.24, its full Update 1.24, and chapter notation; original author TeX. https://arxiv.org/abs/1809.07200.