Research Problems in Function Theory — Problem 2.36

Definitions and mathematical statement

Let $f$ be a transcendental entire function of finite order $\rho=\limsup_{r\to\infty}\log\log M(r,f)/\log r$, with $0<\rho<1$. Fix $\alpha$ satisfying $\rho<\alpha\le1$ and set \[ E_\alpha(f)=\{r\ge1:\log m_0(r,f)>\cos(\pi\alpha)\log M(r,f)\}, \] where $m_0$ and $M$ are the minimum and maximum of $|f|$ on $|z|=r$, respectively, and $\log0=-\infty$. For a measurable $E\subset[1,\infty)$ its upper ordinary density and lower logarithmic density are \[ \overline d(E)=\limsup_{R\to\infty}\frac{|E\cap[1,R]|}{R}, \qquad \underline d_{\log}(E)=\liminf_{R\to\infty}\frac1{\log R}\int_{E\cap[1,R]}\frac{dt}{t}. \] Besicovitch's bound gives $\overline d(E_\alpha)\ge1-\rho/\alpha$; Barry's logarithmic version gives the same lower bound for $\underline d_{\log}$. Determine the best universal lower bound for the ordinary upper density as a function of $(\rho,\alpha)$. In particular, can its stated bound be strictly improved? Sharpness for logarithmic density alone does not settle this ordinary-density question.

Short English statement

Improve, or prove optimal, the ordinary-density bound for circles on which the minimum modulus is large in the cosine theorem.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2302036 (AMR-022-2036), 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 2.36. Original source indexed by AMR-022-2036. https://arxiv.org/abs/1809.07200.