On generalized Voronovskaja theorem for Bernstein polynomials (Q2859485)

The aim of the paper is to prove the following quantitative version of the Voronovskaja theorem for simultaneous approximation of Bernstein operators:NEWLINENEWLINE\[NEWLINE\begin{aligned} &\left| n\left[(B_nf)^{(m)}(x)-f^{(m)}(x)\right]-\frac12\frac{d^m}{dx^m}\{x(1-x)f''(x)\}\right|\leq\\NEWLINE&\leq C(m)\left\{\frac mn\max_{m\leq k\leq m+2}\{| f^{(k)}(x)|\}+(\delta_{n.m}(x))^2\omega_1^ {\varphi}\left(f^{(m+2)},\frac1{\sqrt{n}}\right)\right\},\end{aligned}NEWLINE\]NEWLINENEWLINEwhere \(B_n\) denotes the Bernstein operator of degree \(n\), \(\delta_{n.m}(x)=\varphi(x)+\sqrt{m/n}\), \(\varphi(x)=\sqrt{x(1-x)}\), \(x\in[0,1]\), \(n,m\in\mathbb N\) and \(\omega_1^{\varphi}\) is the first-order Ditzian-Totik modulus of continuity. This result improves in a certain sense an earlier result of \textit{H. Gonska} and \textit{I. Raşa} [Mat. Vesn. 61, No. 1, 53--60 (2009; Zbl 1274.41036)].