Eremenko 1994 extremal problem polynomials

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theorem_1: Eremenko and Lempert's sharp bound on the derivative at a point of modulus one of a monic degree n polynomial whose set of modulus at most one is connected, with equality only for rotations of a shifted Chebyshev polynomial.

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A. Eremenko and L. Lempert, An extremal problem for polynomials, Proc. Amer. Math. Soc. 122 (1994), no. 1, 191--193. Received 18 November 1992, revised 10 December 1992; dedicated to Paul Erdős on his 80th anniversary; 1991 MSC 30C10, 30A10.

The copy read for this card is a publisher scan of the journal version, the three printed pages (head "PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, Volume 122, Number 1, September 1994"; physical PDF p. $n$ is printed p. $190+n$) with a text layer whose formulas are garbled, so the statements below were checked on the page images. Provenance: downloaded in September 2026; the download URL was not recorded; 97,226 bytes. The journal version is the only version read. The scan prints "© 1994 American Mathematical Society" on its first page, every other right reserved.

Reading depth is claims checked: the abstract, properties (i)--(iv) of $f_n$ (p. 191) and Theorem 1 (p. 192) were read clause by clause on the page images; the characterization of $f_n$ (pp. 191--192) and the proof of Theorem 1 (pp. 192--193) were read on the page images but not checked step by step, and are not verified here.

Contents

• The question (p. 191): for a polynomial $f(z)\sim z^n$ as $z\to\infty$ (the paper's (1), so $f$ is monic) with $E_f=\{z:|f(z)|\le1\}$ connected, is $\max\{|f'(z)|:z\in E_f\}\le\tfrac12n^2$ (the paper's (2))? The paper cites Hayman's problem collection (its [1], Problem 4.8), Pommerenke's bound with $en^2/2$ in place of $n^2/2$ (its [2]) and Erdős's survey (its [3]), which observed that the bound in (2) must be relaxed to $\tfrac12\{1+o(1)\}n^2$ and proposed Chebyshev polynomials as the extremal case.
• Extremal polynomials (pp. 191--192): $f_n(z)=T_n(2^{(1-n)/n}z+1)$ with $T_n$ the Chebyshev polynomial; $f_n(0)=1$, $f_n'(0)=2^{(1/n)-1}n^2$, $f_n$ is real with all zeros negative, and its critical values are $\pm1$. The paper states that the normalization (1) together with $f_n(0)=1$, real negative zeros and critical values $\pm1$ (its (1), (i), (iii) and (iv), p. 191) characterizes $f_n$ uniquely; the derivative value (ii) is not among the characterizing properties.
• Theorem 1 (p. 192): let $f$ be a polynomial satisfying (1), $|f(0)|=1$, and $E_f$ connected. Then $|f'(0)|\le f_n'(0)=2^{(1/n)-1}n^2$ (printed as $f_n(0)$, a misprint, since $f_n(0)=1$ by (i)). Equality can occur only for $f(z)=c^{-n}f_n(cz)$, $|c|=1$. Since the hypotheses and the conclusion are invariant under $f(z)\mapsto f(z+c)$, the theorem gives $\max\{|f'(z)|:z\in E_f\}\le2^{(1/n)-1}n^2$ for every monic $f$ of degree $n$ with $E_f$ connected, and the abstract states that this estimate is the best possible.
• Proof (pp. 192--193): an extremal polynomial exists; replacing its zeros $z_k$ by $-|z_k|$ gives a real polynomial $f^*$ with negative zeros that is again extremal and has $E_{f^*}$ connected by the minimum principle; if fewer than $n-1$ critical points of $f^*$ have critical value $\pm1$, the perturbation $f^*(z)+\varepsilon zp(z)$ for a suitable real $p$ keeps the set connected and increases the derivative at $0$, contradicting extremality, so $f^*=f_n$.

Compiled scope

The abstract, properties (i)--(iv) of $f_n$ and Theorem 1 were checked on the page images. The characterization of $f_n$ on pp. 191--192 and the proof on pp. 192--193 were read on the page images but not checked step by step. Nothing here is independently reviewed.

Results. Theorem 1 (p. 192), with the extremal polynomials $f_n$ (p. 191) and the form of the bound for the whole set $E_f$ (abstract and p. 192).

Bears on. #115: Theorem 1, through the paper's translation remark, gives $\max_{E_f}|f'|\le2^{(1/n)-1}n^2=(\tfrac12+o(1))n^2$ for monic $f$ of degree $n$ with $E_f$ connected, attained by $f_n$, which answers yes the problem page's corrected Statement, whose monic hypothesis is the paper's normalization (1); the site's wording, which puts no normalization on the polynomial, is not what the theorem addresses. The value $f_n'(0)$ also shows that the exact bound $\tfrac12n^2$ fails for every $n$.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.