Improvement of one inequality for algebraic polynomials

  • A. N. Nesterenko
  • T. D. Tymoshkevych
  • A. V. Chaikovs'kyi Київ. нац. ун-т iм. Т. Шевченка


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.
How to Cite
Nesterenko, A. N., T. D. Tymoshkevych, and A. V. Chaikovs’kyi. “Improvement of One Inequality for Algebraic Polynomials”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, no. 2, Feb. 2009, pp. 231-42,
Research articles