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Statement

Setting (p. 74). For the five Problems 4.7--4.11 the book writes $E_f^{(n)}=\{z\in\mathbb{C}:|f(z)|\le1\}$, where $n$ is the degree of the polynomial $f$; the first of them, Problem 4.7, takes $f(z)=z^n+a_1z^{n-1}+\ldots+a_n$.

Problem 4.8 (p. 74, quoted). "Assume that $E_f^{(n)}$ is connected. Is it true that

$$\max_{z\in E_f^{(n)}}|f'(z)|\le\frac12n^2\,?\qquad(4.1)$$

Pommerenke [641] proved this with $\frac12en^2$ instead of $\frac12n^2$."

The book's [641] is Ch. Pommerenke, On the derivative of a polynomial, Michigan Math. J. 6 (1959), 373--375. The problem carries no attribution line, and Table 2 (p. 253) lists it among the problems of the 1967 edition.

Update 4.8 (p. 74). The update reports Eremenko's remark that (4.1) is false as stated, Chebyshev polynomials violating it, and that the correct inequality, best possible, is

$$\max_{z\in E_f^{(n)}}|f'(z)|\le2^{1/n-1}n^2,$$

proved by Eremenko and Lempert (the book's [246], printed as "A Eremenko and L. Lempert. An extremal problem for polynomials. 122, 09 1994.", without a journal). It adds a generalisation by Eremenko (the book's [242]).

Source. W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, arXiv:1809.07200v2 (21 September 2018), Chapter 4, p. 74. The edition read is identified on the source card.

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

Proof pointer

None; a problem. Eremenko and Lempert's theorem is on the Eremenko–Lempert Theorem 1 page.

Dependencies

None.

Bears on

• Problem 115: Problem 4.8 is the question with the bound $\frac12n^2$, for the monic polynomials of Problem 4.7; #115 asks for $(\frac12+o(1))n^2$. The sharp bound $2^{1/n-1}n^2$ that Update 4.8 credits to Eremenko and Lempert equals $(\frac12+o(1))n^2$, while the update's Chebyshev remark says the exact bound $\frac12n^2$ fails. The problem page cites this problem for the monic normalization.