Research Problems in Function Theory — Problem 1.11

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

Statement recorded in the supplied source.

Suppose $\rho(f)<\infty$ and $\delta(a,f)>0$ for at least one $a\in\mathbb C$. Must \[ \text{for every }\sigma>1,\qquad \limsup_{r\to\infty}\frac{T(\sigma r,f)}{T(r,f)}<\infty \] hold? This expands the conclusion cross-referenced as Problem 1.10. [S2]

Short English statement

Does one finite deficient value force every fixed radial dilation to increase the characteristic by a bounded factor? The source reports Kotman’s negative answer. [S2]

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2301011 (AMR-022-1011). 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 1.11, its full Update 1.11, and chapter notation; original author TeX. https://arxiv.org/abs/1809.07200.