@article{Nesterenko_Tymoshkevych_Chaikovs’kyi_2009, title={Improvement of one inequality for algebraic polynomials}, volume={61}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/3015}, abstractNote={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.}, number={2}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={Nesterenko, A. N. and Tymoshkevych, T. D. and Chaikovs’kyi, A. V.}, year={2009}, month={Feb.}, pages={231-242} }