TY - JOUR
AU - A. N. Nesterenko
AU - T. D. Tymoshkevych
AU - A. V. Chaikovs'kyi
PY - 2009/02/25
Y2 - 2024/02/22
TI - Improvement of one inequality for algebraic polynomials
JF - Ukrains’kyi Matematychnyi Zhurnal
JA - Ukr. Mat. Zhurn.
VL - 61
IS - 2
SE - Research articles
DO -
UR - https://umj.imath.kiev.ua/index.php/umj/article/view/3015
AB - 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.
ER -