An exact inequality for the derivative of a trigonometric polynomial having only real zeros

Бабенко, В. Ф.; Пичугов, С. А.

doi:10.1007/bf01170244

Cited by 7 publications

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“…logarithmic) asymptotical equality of the quantities on the two sides of (6). However, even the same result as in (6) holds true (perhaps with the unimportant loss of the factor 2) even for complex compacta. We will use such type of estimations in the following form.…”

Section: Chebyshev Estimates and Transfinite Diametersupporting

confidence: 53%

“…Regarding Theorem B, Turán pointed out by example of (1 − x 2 ) n that the √ n order is sharp. Some slightly improved constants can be found in [6] and [31], however, the exact value of the constants and the corresponding extremal polynomials were already computed for all fixed n by Erőd in [21]. Now we are going to describe results concerning Turán-type inequalities (2) for general convex sets.…”

Section: The Oscillation Of a Polynomial In Maximum Normmentioning

confidence: 99%

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Turán type oscillation inequalities in $L^q$ norm on the boundary of convex domains

Glazyrina¹,

Révész²

2017

Mathematical Inequalities &Amp; Applications

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Abstract. Some 77 years ago P. Turán was the first to establish lower estimations of the ratio of the maximum norm of the derivatives of polynomials and the maximum norm of the polynomials themselves on the interval I := [−1,1] and on the unit disk D := {z ∈ C : |z| 1} under the normalization condition that the zeroes of the polynomial p all lie in the interval or in the disk, respectively. He proved that with n := deg p tending to infinity, the precise growth order of the minimal possible ratio of the derivative norm and the norm is √ n for I and n for D . J. Erőd continued the work of Turán and extended his results to several other domains. The growth of the minimal possible ratio of the ∞ -norm of the derivative and the polynomial itself was proved to be of order n for all compact convex domains a decade ago.Although Turán himself gave comments about the above oscillation question in L q norms, till recently results were known only for D and I . Here we prove that in L q norm the oscillation order is again n for a certain class of convex domains, including all smooth convex domains and also convex polygonal domains having no acute angles at their vertices.Mathematics subject classification (2010): 41A17, 30E10, 52A10.

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Section: Chebyshev Estimates and Transfinite Diametersupporting

confidence: 53%

Section: The Oscillation Of a Polynomial In Maximum Normmentioning

confidence: 99%

Turán type oscillation inequalities in $L^q$ norm on the boundary of convex domains

Glazyrina¹,

Révész²

2017

Mathematical Inequalities &Amp; Applications

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“…Inequality (3) of Theorem A is best possible. Regarding (4), Turán pointed out by example of (1−x 2 ) n that the √ n order cannot be improved upon, even if the constant is not sharp, see also [4,19]. The precise value of the constants and the extremal polynomials were computed for all fixed n by Erőd in [14].…”

mentioning

confidence: 99%

“…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%

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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“…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%

A Turán type inequality for rational functions with prescribed poles

Zhou

2005

Acta Math Hung

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An exact inequality for the derivative of a trigonometric polynomial having only real zeros

Cited by 7 publications

References 0 publications

Turán type oscillation inequalities in $L^q$ norm on the boundary of convex domains

Turán type oscillation inequalities in $L^q$ norm on the boundary of convex domains

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

A Turán type inequality for rational functions with prescribed poles

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