On the constants in Videnskiĭ type inequalities for Bernstein operators (Q1682583)

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.