Improvement of one inequality for algebraic polynomials - Nesterenko, A. N.; Tymoshkevych, T. D.; Chaikovs'kyi, A. V. (Ukrains’kyi Matematychnyi Zhurnal, 2009)
We prove that the inequality ||g(⋅/n)||_L_1[−1,1]||P_n+k||_L_1[−1,1] ≤ 2||gP_n+k||_L_1[−1,1], where g : [-1, 1]→ℝ is a monotone odd function and P_n+k is an algebraic polynomial of degree not higher than n + k, is true for all natural n for k = 0 and all natural n ≥ 2 for k = 1. We also propose some other new pairs (n, k) for which this inequality holds. Some conditions on the polynomial P_n+k under which this inequality turns into the equality are established. Some generalizations of this inequality are proposed.