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Source. Theorem 1, p. 1169, of P. Erdős, "Some remarks on polynomials," Bull. Amer. Math. Soc. 53 (1947), 1169-1176. Pages are the journal's own, as on the source card.

Setting

Let $f_n(x)=\prod_{i=1}^n(x-x_i)$ with $-1\le x_1\le x_2\le\cdots\le x_n\le1$, and let $-1\le y_1\le\cdots\le y_{n-1}\le1$ be the roots of $f_n'(x)$ (p. 1169).

Statement

Theorem 1 (p. 1169). For all $n$,

$$\lvert f_n(-1)\rvert+\lvert f_n(+1)\rvert+ \sum_{i=1}^{n-1}\lvert f_n(y_i)\rvert\le 2^n. \tag{1}$$

For $n\ge3$,

$$\lvert f_n(-1)\rvert^{1/2}+\lvert f_n(+1)\rvert^{1/2}+ \sum_{i=1}^{n-1}\lvert f_n(y_i)\rvert^{1/2}\le 2^{n/2}. \tag{2}$$

For $n\ge n_0(k)$,

$$\lvert f_n(-1)\rvert^{1/k}+\lvert f_n(+1)\rvert^{1/k}+ \sum_{i=1}^{n-1}\lvert f_n(y_i)\rvert^{1/k}\le 2^{n/k}. \tag{3}$$

The paper remarks (p. 1169) that if $y_i=y_{i+1}$, or $y_1=-1$, or $y_{n-1}=+1$, the corresponding summands vanish. It shows (p. 1170) that (2) fails for $n<3$, giving $f_1(x)=x$ and, as printed, $f_2(x)=x^2/2-1$, which is not monic; the monic $x^2-\tfrac12$ does violate (2) for $n=2$. It states that equality in (1) and (2) occurs only for $\pm(1\pm x)^n$ (printed with a capital $X$), and that it cannot determine the exact value of $n_0(k)$.

Read depth. Claims checked: the statement, the remarks and the proofs of (1) and (2) were read on the print; the proof of (3) is only sketched in the print.

Proof pointer

Page 1170. For (1), each of $\lvert f_n(-1)\rvert$, $\lvert f_n(y_i)\rvert$ and $\lvert f_n(+1)\rvert$ is at most $2^{n-1}$ times the length of a subinterval of $[-1,1]$, and these lengths sum to at most $2$. For (2), the inequality of the arithmetic and geometric means bounds each square root by $2^{n/2-1}$ times half the sum of two such lengths. For (3) the paper gives only a sketch: for a polynomial maximizing the sum in (3), moving any root lying near $1$ to $-1$ increases the sum, so all roots lie at $-1$.