A new generalization of the Ostrowski inequality and applications

Abstract

A new generalization of the Ostrowski inequality for functions in L p-spaces is introduced and then applied to provide some estimates for the error value of numerical quadrature rules of equal coefficients type.

An analogue of the Ostrowski inequality and applications

“…Proof. The proof of (25) is similar to that of corollary 6 if one replaces x = b in respectively (10) and (12) and then combines them together.…”

“…See also [1,8,11,15] and [13,14,16,19] in this regard. Moreover, the Ostrowski inequality has an important role in numerical quadrature rules [9,12].…”

“…If α(x) ≤ f ′ (x) ≤ β(x) for any x ∈ [a, b] and α, β ∈ C[a, b] then by replacing x = b ∈ [a, b] in (5), the the error of nonstandard quadrature I 6 ( f ) can be bounded as∫ b a (t + a − 2b) β(t) dt ≤ (b − a) ( 2 f (a) − f (b) a − 2b) α(t) dt .(22)Proof. Again, to prove (22) we need to use the results of both theorems 2 and 3 simultaneously such that by replacing x = a in (10) we first obtain(b − a) ( 2 f (a) − f (b) a − 2b) α(t) dt ,(23)provided thatα( t) ≤ f ′ (t) ∀t ∈ [a, b].On the other hand, replacing x = a in(12) gives∫ b a (t + a − 2b) β(t) dt ≤ (b − a) ( 2 f (a) − f (b) ) − ∫ b a f (t) dt ,(24)provided that f ′ (t) ≤ β(t) ∀t ∈ [a, b]. Therefore, combining two latter results (23) and (24) yields (22).…”

Some perturbed Ostrowski type inequalities for functions of bounded variation

“…In order to extend the classical Ostrowski's inequality for di¤erentiable functions with bounded derivatives to the larger class of functions of bounded variation, the author obtained in 1999 (see [17] or the RGMIA preprint version of [19]) the following result for which the constant 1 2 is also sharp. For recent related results, see [1]- [4], [6]- [10], [13]- [15], [26]- [30] and [32]- [44]. then V is also Lipschitzian with the same constant.…”

Inequalities for the Riemann-Stieltjes Integral of Product Integrators With Applications

“…For related results, see [1]- [11], [16]- [17], [21], [23], [25]- [27], [31], [36]- [38], [42], [46]- [52] and [56]- [59].…”