Research Problems in Function Theory — Problem 1.39

Definitions and mathematical statement

For a nonconstant meromorphic function $f$, let $n(t,a)$ count solutions of $f(z)=a$ in $|z|\le t$, with multiplicity; $a=\infty$ means poles. Put \[ N(r,a)=\int_0^r\frac{n(t,a)-n(0,a)}t\,dt+n(0,a)\log r, \quad T(r,f)=N(r,\infty)+\frac1{2\pi}\int_0^{2\pi}\log^+|f(re^{i\theta})|\,d\theta, \] where $\log^+x=\max(0,\log x)$. At a boundary radius $R\in\{1,\infty\}$ with $T(r,f)\to\infty$, define the deficiency $\delta(a,f)=1-\limsup_{r\to R}N(r,a)/T(r,f)$. A deficient value has positive deficiency. Work in the unit disc $\mathbb D=\{z:|z|<1\}$ and define \[ \alpha(f)=\limsup_{r\uparrow1}\frac{T(r,f)}{\log(1/(1-r))}. \] Consider only functions for which this number is finite. Determine the two sharp suprema of $\alpha(f)$ under the following respective hypotheses:

1. $f$ is holomorphic and nowhere zero in $\mathbb D$, and $f'(z)\ne1$ everywhere;
2. $f$ is meromorphic and has no zeros in $\mathbb D$, and $f'(z)\ne1$ wherever the derivative is finite.

In the second class poles are allowed. The source gives universal upper bounds $2$ for the first class and $7$ for the second. It asks whether $2$ is optimal and what the optimal bound in the meromorphic class is. Constants can be included without changing a positive sharp supremum. The finite-growth assumption is retained rather than inferred from the omitted-value conditions.

Short English statement

Find the largest possible logarithmic growth for zero-free disc functions whose derivative omits one, distinguishing holomorphic from meromorphic functions.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2301039 (AMR-022-1039), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 1.39. Original source indexed by AMR-022-1039. https://arxiv.org/abs/1809.07200.