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Statement

Notation (p. 22). Throughout Chapter 2, $f$ is entire and $M(r)=M(r,f)=\max_{|z|=r}|f(z)|$.

Problem 2.16 (p. 29, quoted). "Let $\nu(r)$ be the number of points on $|z|=r$, such that $|f(z)|=M(r,f)$. Can we have

(a) $\limsup_{r\to\infty}\nu(r)=\infty$?

(b) $\liminf_{r\to\infty}\nu(r)=\infty$?"

The book attributes the problem to P. Erdős, and Table 2 (p. 253) lists it among the problems of the 1967 edition. It does not exclude monomials, for which every point of the circle attains the maximum.

Update 2.16 (p. 29). The update credits Herzog and Piranian (the book's [440]: F. Herzog and G. Piranian, The counting function for points of maximum modulus, in Entire Functions and Related Parts of Analysis, Proc. Sympos. Pure Math., La Jolla, 1966, Amer. Math. Soc., 1968, 240--243) with showing that (a) is possible, and records the answer to (b) as still unknown. It adds that they also gave a univalent function in $\mathbb{D}$ for which the analogue of (a) holds.

Source. W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, arXiv:1809.07200v2 (21 September 2018), Chapter 2, p. 29. The edition read is identified on the source card.

Read depth. Claims checked: the notation, the problem, its update and the cited reference entry were read clause by clause on the printed pages. The book proves nothing; it poses and reports.

Proof pointer

None; a problem.

Dependencies

None.

Bears on

• Problem 1117: the same two questions, which the site states for entire functions that are not monomials. As of 2018, Update 2.16 credits Herzog and Piranian with an affirmative answer to (a) and records (b) as unknown; the claim page Herzog and Piranian cites it for (a).