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Source. Question 1.1, p. 2, of Adi Glücksam and Leticia Pardo-Simón, An approximate solution to Erdős' maximum modulus points problem, J. Math. Anal. Appl. 531 (2024), no. 1, Paper No. 127768, DOI 10.1016/j.jmaa.2023.127768, the edition named on the source card. Labels and page numbers are those of the arXiv version arXiv:2208.11154v2 (26 September 2023).

Statement

Setting (p. 1). For an entire $f$, $M(r)=\max_{|z|=r}|f(z)|$, and a maximum modulus point is a point $z$ with $|f(z)|=M(|z|)$. Unless $f$ is a monomial, each circle $\{|z|=r\}$, $r>0$, contains only finitely many of them, and $v(r)=v_f(r)$ is their number.

Question 1.1 (Erdős; p. 2). Let $f$ be a non-monomial entire function.

• (a) Can $v$ be unbounded, that is, can $\limsup_{r\to\infty}v(r)=\infty$?
• (b) Can $v$ tend to infinity, that is, can $\liminf_{r\to\infty}v(r)=\infty$?

The paper dates the question to 1964 and traces it (p. 2) to Hayman's 1967 problem book [Hay67] and its 50th anniversary edition [HL19, Problem 2.16]; part (b) is also attributed to Clunie in [ABB77, Problem 2.49] and appears as [HL19, Problem 2.49]. It records that Herzog and Piranian [HP68] answered (a) positively with an entire $f$ having $v(n)=n$ for each $n\in\mathbb N$, that their refinements do not seem to give any control of $v(r)$ for $r\notin\mathbb N$, and that (b) was unanswered when the paper was written (p. 2).

Read depth. Claims checked: the question, the definition of $v(r)$ and the history on pp. 1--2 were read clause by clause.

Proof pointer

None. The paper poses the question as background and does not answer it; its Theorem 1.2 is an approximate version of (b).

Dependencies

None.

Bears on

• Problem 1117: Question 1.1 is the problem's two questions, with $v(r)$ in place of the problem's $\nu(r)$; part (a) is the problem's first question and part (b) its second.