Research Problems in Function Theory — Problem 5.71

Definitions and mathematical statement

Let $f(z)=\sum_{k\ge1}a_kz^{n_k}$ be holomorphic in the unit disc, where the nonzero exponents are positive integers satisfying $n_{k+1}/n_k\ge q>1$ for a fixed $q$. Define \[ T(r,f)=\frac1{2\pi}\int_0^{2\pi}\log^+|f(re^{i\theta})|\,d\theta, \quad m(r,w;f)=\frac1{2\pi}\int_0^{2\pi} \log^+\frac1{|f(re^{i\theta})-w|}\,d\theta. \] Assume $T(r,f)\to\infty$ as $r\uparrow1$. The deficiency of a finite value $w\in\mathbb C$ is $\delta(w,f)=\liminf_{r\uparrow1}m(r,w;f)/T(r,f)$. Must $\delta(w,f)=0$ for every finite $w$? The question concerns the relative rarity measured by integrated logarithmic means; merely proving that each value is assumed infinitely often is weaker. The gap condition holds with one ratio $q$ for the complete exponent sequence. The coefficients may tend to zero, provided their combined growth still makes $T$ unbounded. Primary work of Tsuboi gives additional sufficient coefficient-growth conditions and recalls Murai's result when the coefficients do not tend to zero. Those special cases do not remove the general unbounded-characteristic hypothesis posed here.

Short English statement

Does every disc power series with Hadamard gaps and unbounded characteristic have no finite deficient value?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2305071 (AMR-022-5071), 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 5.71. Original source indexed by AMR-022-5071. https://arxiv.org/abs/1809.07200.
• S3 N. Tsuboi, The defects of power series in the unit disk with Hadamard gaps, introduction and sufficient conditions. https://www.cajpn.org/complex/refs/thesis/16D-Tsuboi.pdf.