Inequalities for the derivatives of polynomials with real zeros

Бабенко, В. Ф.; Пичугов, С. А.

doi:10.1007/bf01057288

Cited by 7 publications

(11 citation statements)

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“…Now dist (z, J) = min 3/4≤t≤1 |z − τ | (where τ := te i(π/2−2θ) δ), and by the cosine theorem |z − τ | 2 = r 2 + t 2 δ 2 − 2rtδ cos(π/2 − ϕ − 2θ). Because of cos(π/2 − ϕ − 2θ) = sin(ϕ + 2θ) ≤ sin(3θ) ≤ 3 sin θ, (23) implies |z − τ | 2 = r 2 + 10t 2 δr sin θ − 6tδr sin θ = r 2 + 10t 2 − 6t δr sin θ. and thus min 3…”

Section: A Refined Estimate By Tilting the Normal Linementioning

confidence: 95%

“…This was solved by Zhou in a series of papers [36,37,38,39,40]. For more discussions on these results, as well as related results on the interval, period and circle, see the detailed survey in [15] and the introduction of [16], as well as the original works of Babenko and Pichugov [3], Bojanov [7], Varma [32] Babenko et al [3,4] Bojanov [8] and Tyrygin [29,30,18]; see also [31,33,34,18].…”

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confidence: 99%

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Turán—Erőd Type Converse Markov Inequalities on General Convex Domains of the Plane in the Boundary Lq Norm

Glazyrina

Révész

2018

Proc. Steklov Inst. Math.

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In 1939 P. Turán started to derive lower estimations on the norm of the derivatives of polynomials of (maximum) norm 1 on I := [−1, 1] (interval) and D := {z ∈ C : |z| ≤ 1} (disk), under the normalization condition that the zeroes of the polynomial in question all lie in I or D, respectively. For the maximum norm he found that with n := deg p tending to infinity, the precise growth order of the minimal possible derivative norm is √ n for I and n for D. J. Erőd continued the work of Turán considering other domains. Finally, a decade ago the growth of the minimal possible ∞-norm of the derivative was proved to be of order n for all compact convex domains.Although Turán himself gave comments about the above oscillation question in L q norms, till recently results were known only for D and I. Recently, we have found order n lower estimations for several general classes of compact convex domains, and conjectured that even for arbitrary convex domains the growth order of this quantity should be n. Now we prove that in L q norm the oscillation order is at least n/ log n for all compact convex domains.Dedicated to Sergey V. Konyagin on the occasion of his sixtieth birthday MSC 2000 Subject Classification. Primary 41A17. Secondary 30E10, 52A10.

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Section: A Refined Estimate By Tilting the Normal Linementioning

confidence: 95%

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confidence: 99%

Turán—Erőd Type Converse Markov Inequalities on General Convex Domains of the Plane in the Boundary Lq Norm

Glazyrina

Révész

2018

Proc. Steklov Inst. Math.

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“…On the straight part L, where (i) holds, the situation is even simpler as (i) directly entails L |p ′ | q ≥ δ sin θ 8∆d q n q L |p| q . 4 Here, and throughout this section, (i) and (ii) refer to the respective properties in Lemma 5.…”

Section: Completion Of the Proof Of Theoremmentioning

confidence: 99%

“…The best possible constants in some cases were found by Babenko and Pichugov [4], Bojanov [8] and Varma [31]. The classical inequalities of Bernstein and Markov are generalized to various differential operators, too, see [12,21,2,24,3,11,17,18].…”

Section: Introductionmentioning

confidence: 99%

Turán type converse Markov inequalities in Lq on a generalized Erőd class of convex domains

Glazyrina

Révész

2017

Journal of Approximation Theory

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Abstract. P. Turán was the first to derive lower estimations on the uniform norm of the derivatives of polynomials p of uniform norm 1 on the interval I := [−1, 1] and the disk D := {z ∈ C : |z| ≤ 1}, under the normalization condition that the zeroes of the polynomial p in question all lie in I or D, resp. Namely, in 1939 he proved that with n := deg p tending to infinity, the precise growth order of the minimal possible derivative norm is √ n for I and n for D. Already the same year J. Erőd considered the problem on other domains. In his most general formulation, he extended Turán's order n result on D to a certain general class of piecewise smooth convex domains. Finally, a decade ago the growth order of the minimal possible norm of the derivative was proved to be n for all compact convex domains.Turán himself gave comments about the above oscillation question in L q norm on D. Nevertheless, till recently results were known only for I, D and so-called Rcircular domains. Continuing our recent work, also here we investigate the Turán-Erőd problem on general classes of domains.

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“…This inequality was studied quite extensively (interested readers could find useful information in the survey paper [4]). It was generalized to L p spaces, 0 < p < ∞ (see [7][8][9][12][13][14]), and its optimal constants were estimated (see [1][2][3][7][8][9]). Furthermore, the weighted Turán type inequality is also investigated (see [10][11]).…”

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confidence: 99%