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Source. Theorem 7, p. 1175, and the Corollary, p. 1176, 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.

Statement

Theorem 7 (p. 1175). Let $f_n(z)$ be a polynomial of degree $n$ with real coefficients and $\lvert f_n(z)\rvert<1$ for $-1\le z\le1$. Then for $\lvert z_0\rvert\ge1$,

$$\lvert f_n(z_0)\rvert\le\lvert T_n(z_0)\rvert ,$$

where $T_n$ is the Chebyshev polynomial of degree $n$. The print adds that equality holds only for $f_n(z)=\pm T(z)$, although $T_n$ itself does not satisfy the strict hypothesis. It notes that for real $z_0$ the result is well known.

The paper proves a more general form, (12) (p. 1175): the same inequality holds for $\lvert z_0\rvert\ge1$ when $\lvert f_n\rvert\le1$ is assumed only at the $n+1$ points $-1$, $1$ and the roots of $T_n'$.

Corollary (p. 1176). If $\lvert f_n(z)\rvert\le1$ for $-1\le z\le1$ and $f_n$ has real coefficients, then $\lvert f_n(z)\rvert<\lvert T_n(i)\rvert$ for $\lvert z\rvert\le1$.

The paper remarks (p. 1176) that without real coefficients the corollary can fail, that it cannot determine $\max\lvert f_n(z)\rvert$ for $\lvert z\rvert\le1$ in that case, and that the same method shows that $\sum_k\lvert a_k\rvert$ is maximal for $f=\pm T_n$ among real polynomials bounded by $1$ on $[-1,1]$; it quotes Szegő's stronger statement that $\lvert a_{2k}\rvert+\lvert a_{2k+1}\rvert$ is maximal for $f=\pm T_n$.

Read depth. Claims checked: the statements were read on the print. The proof (pp. 1175-1176) was read but not checked step by step.

Proof pointer

Pages 1175-1176. Write $f_n(z_0)=\sum_i y_il_i(z_0)$ by Lagrange interpolation at the extremal points of $T_n$, with real $y_i$ of absolute value at most $1$. A geometric argument shows that the vectors $(-1)^il_i(z_0)$ pairwise make angles less than $\pi/2$, because $[-1,1]$ subtends an angle at most $\pi/2$ from $z_0$; so $\lvert f_n(z_0)\rvert$ is largest when $y_i=\pm(-1)^i$, which gives $\pm T_n(z_0)$.