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 nn points and mm-th derivative. To be precise, the following inequality equation(⋆⋆)|1b−a∫abf(x)dx−b−an∑i=1nf(a+xi(b−a))|≦2m+54(b−a)m+1(m+1)!(S−s) holds, where S≔supa≦x≦bf(m)(x)S≔supa≦x≦bf(m)(x), s≔infa≦x≦bf(m)(x)s≔infa≦x≦bf(m)(x) and for suitable x1,x2,…,xnx1,x2,…,xn. It is worth noticing that nn, mm are arbitrary numbers. This means that the estimate in (⋆)(⋆) is more accurate when mm is large enough. Our approach is also elementary.