A new Ostrowski–Grüss inequality involving 3n knots

摘要

This is the fifth and last in our series of notes concerning some classical inequalities such as the Ostrowski, Simpson, Iyengar, and Ostrowski–Grüss inequalities in R. In the last note, we propose an improvement of the Ostrowski–Grüss inequality which involves 3n knots where n≧1 is an arbitrary numbers. More precisely, suppose that {xk}k=1n⊂[0,1],{yk}k=1n⊂[0,1], and {αk}k=1n⊂[0,n] are arbitrary sequences with ∑k=1nαk=n and ∑k=1nαkxk=n/2. The main result of the present paper is to estimate1n∑k=1nαkfa+(b-a)yk-1b-a∫abf(t)dt-f(b)-f(a)n∑k=1nαkyk-xkin terms of either f′ or f″. Unlike the standard Ostrowski–Grüss inequality and its known variants which basically estimate f(x)-∫abf(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 {xk}k=1n,{yk}k=1n and {αk}k=1n. As far as we know, this is the first result involving the Ostrowski–Grüss inequality with three sequences of parameters.