Research Problems in Function Theory — Problem 1.35

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

An algebraic ordinary differential equation of order $n\ge1$ is \[ P(z,f(z),f'(z),\ldots,f^{(n)}(z))=0, \] with a nonzero polynomial $P\in\mathbb C[z,w_0,\ldots,w_n]$ depending on $w_n$. Polynomial coefficients include rational coefficients after clearing denominators. Put $\exp_1(t)=e^t$ and $\exp_{j+1}(t)=\exp(\exp_j(t))$.

Statement recorded in the supplied source.

Determine upper and lower growth estimates for entire and meromorphic solutions of algebraic ordinary differential equations, including dependence on the order $n$ of the equation. For meromorphic solutions growth is measured by $T(r,f)$; for entire solutions one may use $\log M(r,f)$. In particular, test the source's proposed entire-solution bound \[ \log|f(z)|\le\exp_n(|z|)\quad\text{for all sufficiently large }|z|, \] equivalently $\log M(r,f)\le\exp_n(r)$ for all sufficiently large $r$. [S2]

Short English statement

Determine growth bounds for algebraic differential-equation solutions in both classes and test the proposed iterated-exponential bound for entire solutions. The source gives partial results rather than a general resolution. [S2]

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2301035 (AMR-022-1035). 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 1.35, its full Update 1.35, and chapter notation; original author TeX. https://arxiv.org/abs/1809.07200.