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Polynomial problem with inequality

Source: Latvian TST for Baltic Way 2019 Problem 4

May 29, 2020
algebrapolynomialinequalitieschebyshev polynomial

Problem Statement

Let P(x)P(x) be a polynomial with degree nn and real coefficients. For all 0≤y≤10 \le y \le 1 holds ∣p(y)∣≤1\mid p(y) \mid \le 1. Prove that p(−1n)≤2n+1−1p(-\frac{1}{n}) \le 2^{n+1} -1
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