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Statement

Setting (p. 191). The paper's (1) is the normalization $f(z)\sim z^n$ as $z\to\infty$, so $f$ is a monic polynomial of degree $n$, and $E_f=\{z:\lvert f(z)\rvert\le1\}$. With $T_n$ the Chebyshev polynomial, $\cos nz=T_n(\cos z)$, the paper sets

$$f_n(z)=T_n\bigl(2^{(1-n)/n}z+1\bigr),\qquad n=1,2,\ldots,$$

and lists four properties of $f_n$: (i) $f_n(0)=1$; (ii) $f_n'(0)=2^{(1/n)-1}n^2$; (iii) $f_n$ is real and all its zeros are negative; (iv) the critical values of $f_n$ are $\pm1$. The paper states (pp. 191--192) that (1), (i), (iii) and (iv) characterize $f_n$ uniquely.

Theorem 1 (p. 192). Let $f$ be a polynomial satisfying (1) with $\lvert f(0)\rvert=1$ and $E_f$ connected. Then

$$\lvert f'(0)\rvert\le2^{(1/n)-1}n^2,$$

and equality can occur only for $f(z)=c^{-n}f_n(cz)$ with $\lvert c\rvert=1$. The print gives the bound as "$\lvert f'(0)\rvert\le f_n(0)=2^{(1/n)-1}n^2$" [sic]; by (i) and (ii) the middle term is $f_n'(0)$. The polynomial $f_n$ satisfies (1), one of its characterizing properties, and by (i), (ii) and (iv) it has $\lvert f_n(0)\rvert=1$, attains the bound, and has $E_{f_n}$ connected (its critical points have critical values $\pm1$, so they lie in $E_{f_n}$; this step is not written out in the paper), so the bound is sharp, as the abstract states.

Form for the whole set (abstract and p. 192). The paper observes that the hypotheses and the conclusion are invariant under $f(z)\mapsto f(z+c)$, $c\in\mathbb C$, and states that the conjecture follows. The abstract states the result in this form: if $f(z)=z^n+\cdots$ and $E=\{z:\lvert f(z)\rvert\le1\}$ is connected, then $\max\{\lvert f'(z)\rvert:z\in E\}\le2^{(1/n)-1}n^2$, and this estimate is the best possible. (The maximum of $\lvert f'\rvert$ over the compact set $E$ is attained on its boundary, where $\lvert f\rvert=1$, so a translation moves that point to $0$; this step is not written out in the paper.)

Since $2^{(1/n)-1}n^2=\tfrac12n^2+\tfrac12(2^{1/n}-1)n^2$, the bound is $(\tfrac12+o(1))n^2$ as $n\to\infty$, and it exceeds $\tfrac12n^2$ for every $n\ge1$ (an observation of this page).

Source. A. Eremenko and L. Lempert, An extremal problem for polynomials, Proc. Amer. Math. Soc. 122 (1994), no. 1, 191--193: the setting and the properties of $f_n$ on p. 191, their characterization on pp. 191--192, Theorem 1 on p. 192 and its proof on pp. 192--193, in the journal version identified on the source card.

Read depth. Claims checked: the abstract, the setting, properties (i)--(iv) and Theorem 1 were read clause by clause on the page images. The characterization of $f_n$ and the proof of Theorem 1 were read on the page images but not checked step by step. Nothing here is independently reviewed.

Proof pointer

Pp. 191--193. Characterization of $f_n$ (pp. 191--192): for $f$ with (1), (i), (iii) and (iv), all zeros of $f$ and $f'$ are real and simple, a real affine change of variable $g(z)=f(az+b)$ puts the zeros in $(-1,1)$ with $g(1)=1$ and $g(-1)=(-1)^n$, and the rational function $g'^2/(1-g^2)$ is then identified as $n^2/(1-z^2)$, whose solution with $g(1)=1$ is $T_n$. Theorem 1 (pp. 192--193): an extremal polynomial exists. Writing it as $\lambda\prod(1-z/z_k)$ with $\lvert\lambda\rvert=1$ and $\prod\lvert z_k\rvert=1$, the paper replaces it by $f^*(z)=\prod(1+z/\lvert z_k\rvert)$, which satisfies (1) and $f^*(0)=1$, has $E_{f^*}$ connected (its zeros lie on a segment inside $E_{f^*}$, then the minimum principle) and has $(f^*)'(0)\ge\lvert f'(0)\rvert$, with equality only when all $z_k$ have one argument. So $f^*$ is extremal and $f(z)=c^{-n}f^*(cz)$. If fewer than $n-1$ critical points of $f^*$ have critical value $\pm1$, a perturbation $f^*+\varepsilon zp$ with a suitable real polynomial $p$ keeps the hypotheses and increases the derivative at $0$, contradicting extremality; so $f^*$ has properties (1), (i), (iii) and (iv), and $f^*=f_n$.

Dependencies

None beyond classical facts: the Chebyshev polynomials, Rolle's theorem and the minimum principle. The paper cites Hayman's Research problems in function theory (1967), Problem 4.8, for the question, Pommerenke (Michigan Math. J. 6 (1959), 373--375) for the earlier bound $en^2/2$, and Erdős's survey Some of my favorite unsolved problems (1990).

Bears on

• Problem 115: the problem's corrected Statement asks whether a monic polynomial $p$ of degree $n$ with $\{\lvert p\rvert\le1\}$ connected has $\lvert p'\rvert\le(\tfrac12+o(1))n^2$ there. The form for the whole set gives the bound $2^{(1/n)-1}n^2$, which is $(\tfrac12+o(1))n^2$, so it answers that question yes; the normalization (1) is the corrected Statement's monic hypothesis. The polynomial $f_n$ shows that the exact bound $\tfrac12n^2$ fails for every $n$.