Research Problems in Function Theory — Problem 2.11

Definitions and mathematical statement

Let $0<\lambda_1<\lambda_2<\cdots$ be integers and let \[ f(z)=a_0+\sum_{n\ge1}a_nz^{\lambda_n},\qquad a_n\ne0, \qquad \sum_{n\ge1}\frac1{\lambda_n}<\infty, \] be a transcendental entire function. Thus the Taylor expansion has Fejér gaps; convergence is required on the whole complex plane, not just in the unit disc. Set $M(r,f)=\max_{|z|=r}|f(z)|$ and $m_0(r,f)=\min_{|z|=r}|f(z)|$. The problem has two assertions. First, can $f$ have a finite asymptotic value: a complex number $a$ and a continuous curve $\gamma:[0,\infty)\to\mathbb C$ such that $|\gamma(t)|\to\infty$ and $f(\gamma(t))\to a$? The proposed answer is no. Second, must \[ \limsup_{r\to\infty}\frac{\log m_0(r,f)}{\log M(r,f)}=1? \] Here $\log m_0=-\infty$ at radii containing a zero; the denominator is positive for sufficiently large $r$. The reverse inequality is automatic from $m_0\le M$. The source records results under stronger gap assumptions; the summability assumption alone is the intended hypothesis in both questions.

Short English statement

Do summable reciprocal Taylor exponents exclude every finite asymptotic value and force near equality of minimum and maximum modulus along some large circles?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2302011 (AMR-022-2011), 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 2.11. Original source indexed by AMR-022-2011. https://arxiv.org/abs/1809.07200.