An improvement of the constant in Videnskiĭ’s inequality for Bernstein polynomials

Abstract

AbstractIn this paper, we deal with improvements on the constant {M_{n}(\gamma)} in the so-called Videnskiĭ inequality\biggl{|}(B_{n}f)(x)-f(x)-\frac{x(1-x)}{2n}f^{\prime\prime}(x)\biggr{|}\leq M_% {n}(\gamma)\frac{x(1-x)}{n}\omega\bigg{(}f^{\prime\prime};\sqrt{\frac{\gamma}{% n}}\biggr{)}for a fixed constant {\gamma\geq 1} and {x\in[0,1]}, where {B_{n}f} is the Bernstein polynomial, {f\in C^{2}[0,1]} and ω is the first order modulus of continuity. Let {M(\gamma)=\sup_{n\in\mathbb{N}}M_{n}(\gamma)}. We prove the Videnskiĭ inequality for arbitrary {\gamma\geq 1}. In particular, we improve the constant {M(2)=0.9} (Gonska and Ra şa [8], 2008) to {M(2)=0.6875}. Finally, we consider {M_{n}(1)} for small values of n.