A new Ostrowski-Grüss inequality involving 3n knots

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.