Abstract:
Let Φ+ be the set of nondecreasing functions φ defined on (0,∞) which admit a representation φ(u)=ψ(lnu), where the function ψ is convex (below) on (−∞,∞). To the class Φ+ belong, for example, the functions lnu, ln+u, up when p>0, and also any function φ which is convex on (0,∞). In this paper it is shown, in particular, that if φ∈Φ+, then for any trigonometric polynomial Tn of order n the following inequality holds for all natural numbers r:
∫2π0φ(|T(r)n(t)|)dt⩽∫2π0φ(nr|Tn(t)|)dt.
This inequality may be considered a generalization of the inequalities of S. N. Bernstein and A. Zygmund.
Bibliography: 16 titles.
\Bibitem{Are81}
\by V.~V.~Arestov
\paper On~integral inequalities for trigonometric polynomials and their derivatives
\jour Math. USSR-Izv.
\yr 1982
\vol 18
\issue 1
\pages 1--17
\mathnet{http://mi.mathnet.ru/eng/im1545}
\crossref{https://doi.org/10.1070/IM1982v018n01ABEH001375}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=607574}
\zmath{https://zbmath.org/?q=an:0538.42001|0517.42001}
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This publication is cited in the following 65 articles:
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