Variation on a Theme of Chebyshev: Sharp Estimates for the Leading Coefficients of Bounded Polynomials

The n-th Chebyshev polynomial of the first kind, Tn, maximizes various functionals on Bn, the unit ball of real polynomials with respect to the uniform norm on [−1,1], see e.g. [3], [18], [20], [37], [41]. The earliest example (1854) is Chebyshev’s inequality [6] for the leading coefficient of Pn ∈ Bn (where Pn(x) = n∑ k=0 ak x k and degree ≤ n): (i) |an | ≤ 2n−1. In 1892 V.A. Markov [16] found analogous sharp estimates for |an−1 | and for |an−2|, and Szegö did likewise for |an−1|+ |an|, as published by Erdös in 1947 [12]. Only recently we have provided in [34] the sharp estimate for |an−2|+ |an−1 | and have announced in [32] the exact upper bound for |an−2 + an−1 + an|. In Theorem 2.1 we solve the encompassing extremal problem of finding the sharp estimates for all possible compositions of the first three leading coefficients an, an−1, an−2 of Pn ∈ Bn and even of Pn ∈ Cn = {Pn: |Pn(cos (n − i)pin) | ≤ 1 for 0 ≤ i ≤ n}, where Cn ⊃ Bn if n ≥ 2. In Theorem 3.1 w