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Statement

Problem 4.14 (pp. 75--76, quoted). "Does there exist a polynomial of the type in Problem 4.13, for which

$$\min_{|z|=1}|P(z)|>C_2\sqrt{n}\qquad(4.3)$$

for every $n$? More generally, does there exist such a polynomial satisfying both (4.2) and (4.3)?"

The type in Problem 4.13 (p. 75) is $P(z)=\sum_{k=1}^n\varepsilon_kz^k$ with each $\varepsilon_k=\mp1$, and (4.2) is the upper bound $\max_{|z|=1}|P(z)|<C_1\sqrt n$; see Problem 4.13. The problem carries no attribution line, and Table 2 (p. 253) lists it among the problems of the 1967 edition.

Update 4.14 (p. 76). The update credits an affirmative answer to Beller and Newman (the book's [79]: E. Beller and D. J. Newman, The minimum modulus of polynomials, Proc. Amer. Math. Soc. 45 (1974), 463--465) for coefficients with $|\varepsilon_k|\le1$, and to Körner (the book's [490]: T. W. Körner, On a polynomial of Byrnes, Bull. London Math. Soc. 12 (1980), 219--224) for coefficients with $|\varepsilon_k|=1$. It records the case $\varepsilon_k=\pm1$ as open.

Observation made here: the corpus's page for Problem 230 records that Bombieri and Bourgain (footnote 1, p. 627 of their paper; see the Bombieri–Bourgain card) say the proofs of Körner's Theorems 6 and 7 rest on an incorrect theorem of Byrnes. The update does not mention this.

Source. W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, arXiv:1809.07200v2 (21 September 2018), Chapter 4, pp. 75--76. The edition read is identified on the source card.

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

Proof pointer

None; a problem. The construction for $\pm1$ coefficients is on the Balister–Bollobás–Morris–Sahasrabudhe–Tiba card, a 2020 paper the 2018 update predates.

Dependencies

Problem 4.13, for the polynomials and the bound (4.2).

Bears on

• Problem 228: the "more generally" question of Problem 4.14, read with $\varepsilon_k=\pm1$, asks for #228's two-sided bound. On $|z|=1$ the book's $P(z)=\sum_{k=1}^n\varepsilon_kz^k$ has the modulus of the degree $n-1$ polynomial $\sum_{k=1}^n\varepsilon_kz^{k-1}$, so the missing constant term shifts the degree by one; the book asks for every $n$, the problem for all large $n$. As of 2018 the update records this case as open; the problem page records the later work.