An extremum problem concerning algebraic polynomials

Erdös, P.; Varma, A. K.

doi:10.1007/bf01949134

Cited by 10 publications

(8 citation statements)

References 4 publications

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“…These problems are related to some previous results due to , Milovanovic [6], Erdös and Varma [2], and also to the classical inequalities of A. Markov [5], P. Erdös [1], G. G. Lorentz [3,4], and G. Szegö [8].…”

Section: Introductionsupporting

confidence: 52%

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Extremal problems for Lorentz classes of nonnegative polynomials in 𝐿² metric with Jacobi weight

Milovanović¹,

Petković²

1988

Proc. Amer. Math. Soc.

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Section: Introductionsupporting

confidence: 52%

“…It is of interest to note that Erdös and Varma [2] proved that the best constant in the Lorentz class Ln (n > 2) for a = ß = 0 is the same one as that in (3.6), i.e. CÍ0)(0,0) = CJ^ÍO.O).…”

Section: Some Auxiliary Resultsmentioning

confidence: 99%

Extremal problems for Lorentz classes of nonnegative polynomials in 𝐿² metric with Jacobi weight

Milovanović¹,

Petković²

1988

Proc. Amer. Math. Soc.

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“…For this function, integral (3) is the length of the graph of the polynomial f n on the period [0, 2π]. In this case, the problem was posed and solved by Erdös in 1939 [8]. For convex nondecreasing functions ϕ, inequality (3) was rediscovered by Taikov [25].…”

Section: Introductionmentioning

confidence: 99%

Integral inequalities for algebraic and trigonometric polynomials

Arestov

Glazyrina

2012

Dokl. Math.

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Abstract. We investigate sharp estimates of integral functionals for operators on the set T n of real trigonometric polynomials f n of degree n ≥ 1 by the uniform norm f n C2π of the polynomials and similar questions for algebraic polynomials on the unit circle of the complex plane. P. Erdös, A.P. Calderon, G. Klein, L.V. Taikov, and others investigated such inequalities. In this paper, we, in particular, show that, for 0 ≤ q < ∞, the sharp inequality D α f n Lq ≤ n α cos t Lq f n ∞ holds on the set T n for the Weyl fractional derivatives D α f n of order α ≥ 1. For q = ∞ (α ≥ 1), this fact was proved by P.I. Lizorkin (1965). For 1 ≤ q < ∞ and positive integer α, the inequality was proved by Taikov (1965); however, in this case, the inequality follows from results of an earlier paper by Calderon and Klein (1951).

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“…Later Erdos [4], Lorentz [8], Erdos and Varma [5] and Szabados and Varma [15] showed that by restricting the form of the polynomials, substantially better bounds for the derivatives can be obtained. A. K. Varma, T. M. Mills and Simon J. Smith [2] In the case of the Markoff inequality, the condition (1.1) ensures that the graph of the polynomial p n {x) is contained in the square -1 < x < 1 , -1 < j < 1.…”

Section: Introductionmentioning

confidence: 99%