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Statement

Problem 4.13 (p. 75, quoted). "It is known that there exists a polynomial $P(z)$

$$P(z)=\sum_{k=1}^{n}\varepsilon_kz^k,\qquad\varepsilon_k=\mp1$$

for which

$$\max_{|z|=1}|P(z)|<C_1\sqrt{n}.\qquad(4.2)$$

(See Clunie [158]). Is it necessarily true that $C_1>1+A$ if (4.2) holds, where $A$ is a positive absolute constant?"

The book's [158] is J. Clunie, On schlicht functions, Ann. of Math. (2) 69 (1959), 511--519. The problem carries no attribution line, and Table 2 (p. 253) lists it among the problems of the 1967 edition.

Update 4.13 (p. 75). No progress had been reported to the authors.

Source. W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, arXiv:1809.07200v2 (21 September 2018), Chapter 4, p. 75. The edition read is identified on the source card.

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

Proof pointer

None; a problem.

Dependencies

None.

Bears on

• Problem 230: Problem 4.13 asks for coefficients $\pm1$ what #230 asks for complex coefficients of modulus one, both with the sum from $k=1$: whether the maximum modulus on the circle is at least $(1+c)\sqrt n$ for an absolute $c>0$; #230 asks this for $n\ge2$, and the book states no range of $n$. The $\pm1$ polynomials lie in #230's class, so an affirmative answer to #230 would answer Problem 4.13 affirmatively for $n\ge2$; a negative answer to #230 does not settle Problem 4.13.