Improvement of one inequality for algebraic polynomials
Abstract
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.
English version (Springer): Ukrainian Mathematical Journal 61 (2009), no. 2, pp 277-291.
Citation Example: Chaikovs'kyi A. V., Nesterenko A. N., Tymoshkevych T. D. Improvement of one inequality for algebraic polynomials // Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 231-242.
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