Research Problems in Function Theory — Problem 2.68

Definitions and mathematical statement

Let $\rho>0$ and let $f$ be an entire function with \[ \log M(r,f)\le(1+o(1))r^\rho\quad(r\to\infty), \qquad M(r,f)=\max_{|z|=r}|f(z)|. \] Suppose a continuous curve $\Gamma$ tends to infinity and, for some fixed $\alpha\in[-1,1)$, satisfies $\log|f(z)|\le(\alpha+o(1))|z|^\rho$ as $z\to\infty$ along $\Gamma$. For $\varepsilon>0$ define the angular measure, in radians, \[ E(r,\varepsilon)=\left|\{\theta\in[0,2\pi): \log|f(re^{i\theta})|\le(1-\varepsilon)r^\rho\}\right|. \] Zeros are included by the convention $\log0=-\infty$. The conjecture is \[ \lim_{\varepsilon\downarrow0}\limsup_{r\to\infty}E(r,\varepsilon) \ge \frac2\rho\arccos\alpha. \] The outer limit exists by monotonicity as $\varepsilon$ decreases. This specifies the joint limiting notation used in the source. The same curve supplies the small-value hypothesis at all sufficiently large points on it; a sequence of unrelated small-value points is insufficient. The proposed conclusion measures how much of some large circles must consequently lie below the near-maximal growth scale.

Short English statement

Does unusually small growth along one curve force a quantitatively large set of small-growth angles on arbitrarily large circles?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2302068 (AMR-022-2068), 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 2.68. Original source indexed by AMR-022-2068. https://arxiv.org/abs/1809.07200.