A new way to think about Ostrowski-like type inequalities

In this present paper, by considering some known inequalities of Ostrowski-like type, we propose a new way to treat a class of Ostrowski-like type inequalities involving n points and m-th derivative. To be precise, the following inequality 1 b 􀀀 a Z b a f .x/ dx 􀀀 b 􀀀 a n Xn iD1 f .a C xi .b 􀀀 a// 5 2m C 5 4 .b 􀀀 a/mC1 .m C 1/W .S 􀀀 s/ (?) holds, where S VD supa5x5b f .m/.x/, s VD infa5x5b f .m/.x/ and for suitable x1; x2; : : : ; xn. It is worth noticing that n, m are arbitrary numbers. This means that the estimate in .?/ is more accurate when m is large enough. Our approach is also elementary.