On the Inequality of I. Schur

Milev, Lozko; Nikolov, Geno

doi:10.1006/jmaa.1997.5658

Cited by 6 publications

(4 citation statements)

References 6 publications

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“…The answer is in the affirmative, as was shown in a recent paper of Milev and Nikolov [11]. Even more turned out to be true.…”

Section: Theorem B If F ∈ S N Thensupporting

confidence: 53%

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An extension of an inequality of I. Schur

Nikolov¹

2005

Mathematische Nachrichten

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Denote by πn the set of all algebraic polynomials of degree at most n with complex coefficients. An inequality of I. Schur asserts that the first derivative of the transformed Tchebycheff polynomial $\overline {T}_n (x) = T_n (x \, \rm {cos} \, {{\pi} \over {2n}})$ has the greatest uniform norm in [−1, 1] among all f ∈ 𝒮n, where equation image Here we show that this extremal property of $\overline {T}_n$ persists in the wider class of polynomials f ∈ πn which vanish at ±1, and for which there exist n − 1 points $\{t_j\}^{n-1}_{j=1}$ separating the zeros of $\overline {T}_n$ and such that $|f(t_j)| \le |\overline {T}_n (t_j)|$ for j = 1, …, n − 1. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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“…The answer is in the affirmative, as was shown in a recent paper of Milev and Nikolov [11]. Even more turned out to be true.…”

Section: Theorem B If F ∈ S N Thensupporting

confidence: 53%

“…is attained only at the end points (see e.g., [11,Lemma 2.4]). This fact follows also from Schur's proof of Theorem B.…”

Section: Moreover Equality Occurs If and Only Ifmentioning

confidence: 99%

An extension of an inequality of I. Schur

Nikolov¹

2005

Mathematische Nachrichten

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“…is known as being the extremal polynomial in the classical Schur inequality [12], which is the analogue of the Markov inequality for polynomials satisfying zero boundary conditions. Precisely, Schur's inequality asserts that T, has the greatest uniform norm of its first derivative on [-1, 1] among allfE 7rn, satisfyingf(+l) 0 and [[f[I < 1. In a recent paper Milev and Nikolov [5] have shown that the extremality of Tn persists for higher order derivatives as well. Precisely, Milev and Nikolov proved the following extension of Schur's inequality.…”

Section: Cos 2nmentioning

confidence: 97%

An inequality for polynomials with elliptic majorant

Nikolov¹

1999

J Inequal Appl

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“…For some inequalities of the Duffin and Schaeffer type in the uniform norm · C[ −1,1] the reader is referred to [3,7,[12][13][14][15][16][17].…”

Section: Definitionmentioning

confidence: 99%

An extremal property of Hermite polynomials

Nikolov

2004

Journal of Mathematical Analysis and Applications

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Let H n be the nth Hermite polynomial, i.e., the nth orthogonal on R polynomial with respect to the weight w(x) = exp(−x 2 ). We prove the following: If f is an arbitrary polynomial of degree at most n, such that |f | |H n | at the zeros of H n+1 , then for k = 1, . . . , n we have f (k) H (k) n , where · is the L 2 (w; R) norm. This result can be viewed as an inequality of the Duffin and Schaeffer type. As corollaries, we obtain a Markov-type inequality in the L 2 (w; R) norm, and estimates for the expansion coefficients in the basis of Hermite polynomials.

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On the Inequality of I. Schur

Cited by 6 publications

References 6 publications

An extension of an inequality of I. Schur

An extension of an inequality of I. Schur

An inequality for polynomials with elliptic majorant

An extremal property of Hermite polynomials

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