A new Ostrowski-Gruss inequality involving 3n knots

作者：Vu Nhat Huy; Quoc Anh Ngo^*

来源：Applied Mathematics and Computation, 2014, 235: 272-282.

DOI：10.1016/j.amc.2014.02.090

• Integral inequality
• Taylor expansion
• Ostrowski
• Ostrowski-Gruss
• Simpson
• Iyengar
• Bernoulli polynomial

This is the fifth and last in our series of notes concerning some classical inequalities such as the Ostrowski, Simpson, Iyengar, and Ostrowski-Gruss inequalities in R. In the last note, we propose an improvement of the Ostrowski-Gruss inequality which involves 3n knots where n >= 1 is an arbitrary numbers. More precisely, suppose that {x(k)}(k-1)(n) subset of [0, 1], {y(k)}(k-1)(n) subset of [0, 1], and {alpha(k)}(k-1)(n) subset of [0, n] are arbitrary sequences with Sigma(n)(k-1) alpha(k) = n and Sigma(n)(k-1) alpha(k)x(k) = n/2. The main result of the present paper is to estimate
1/n Sigma(n)(k=1) alpha(k)f(a + (b - a)y(k)) - 1/b - a integral(b)(a) f(t)dt - f(b) - f(a)/n Sigma(n)(k=1) alpha(k)(y(k) - x(k))
in terms of either f' or f ''. Unlike the standard Ostrowski-Gruss inequality and its known variants which basically estimate f (x) - (integral(b)(a)f(t)dt) / (b - a) in terms of a correction term as a linear polynomial of x and some derivatives of f, our estimate allows us to freely replace f(x) and the correction term by using 3n knots {x(k)}(k-1)(n), {y(k)}(k-1)(n) and {a(k)}(k-1)(n). As far as we know, this is the first result involving the Ostrowski-Gruss inequality with three sequences of parameters.

• 出版日期2014-5-25