On the constants in Videnskiĭ type inequalities for Bernstein operators
type: Item
instance of: scholarly article
title: On the constants in Videnskiĭ type inequalities for Bernstein operators
zbMATH Open document ID: 1381.41018
author: Daniel Cárdenas Morales
published in: Results in Mathematics
publication date: 2017-11-30T00:00:00Z
review text: Let \(B_n\), \(n \geq 1\), be the classical Bernstein operators on \(C[0,1]\). A Videnskiĭ\ type inequality is a quantitative version of the well known Voronovskaja formula and has the form \[ \left| B_n(f;x)-f(x)-\frac{x(1-x)}{2n} f^{\prime\prime}(x) \right| \leq K\, \frac{x(1-x)}{n}\, \omega (f^{\prime\prime}, \sqrt{2/n}), \qquad(1) \] where \(K\) is an absolute constant, \(\omega\) is the usual modulus of continuity, and \(f \in C^2[0,1]\). In 1985, V.S. Videnskiĭ\ proved (1) with \(K=1\). In 2008, H. Gonska and the reviewer showed that \(K=0.9\) is also admissible. Recently, U. Abel and H. Siebert proved that (1) holds true with \(K=11/16=0.6875\). In Section 2 of this paper, the author investigates very carefully the central moments of order \(6\) and \(8\) of \(B_n\), and obtains new upper bounds for them, better than the ones recently obtained by \textit{J. A. Adell}, \textit{J. Bustamante} and \textit{J. M. Quesada} [Appl. Math. Comput. 265, 723--732 (2015)]. Then quantitative Voronovskaja type results are presented, and (1) is proved for \(K=617/1024=0.6025390625\). The best admissible value of \(K\) is not known; in the last section the author presents some aspects concerning this problem.
reviewed by: Ioan Raşa
Mathematics Subject Classification ID: 41A36
Mathematics Subject Classification ID: 41A25
zbMATH DE Number: 6814119
zbMATH Keywords: Bernstein operators
zbMATH Keywords: central moments
zbMATH Keywords: Videnskiĭ's inequality
zbMATH Keywords: Voronovskaja formula
zbMATH Keywords: modulus of continuity
MaRDI profile type: Publication
full work available at URL: s00025-017-0707-3
OpenAlex ID: W2625171629
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DOI: 10.1007/S00025-017-0707-3
description: scientific article; zbMATH DE number 6814119