We prove that the inequality \(\vert\vert g (\cdot / n ) \vert\vert_{L_{1}[-1,1]} \vert\vert P_{n+k}\vert\vert_{L_{1}[-1,1]} \leq 2 \vert\vert gP_{n+k}\vert\vert_{L_{1}[-1,1]}\), 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.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 2, pp. 231–242, February, 2009.
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Nesterenko, O.N., Tymoshkevych, T.D. & Chaikovs’kyi, A.V. Improvement of one inequality for algebraic polynomials. Ukr Math J 61, 277–291 (2009). https://doi.org/10.1007/s11253-009-0205-1
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DOI: https://doi.org/10.1007/s11253-009-0205-1