A sharp companion of Ostrowski’s inequality for the Riemann–Stieltjes integral and applications

Abstract

A sharp companion of Ostrowski’s inequality for the Riemann-Stieltjes integral $\int_a^b {f(t)\;du(t)} $, where f is assumed to be of r-H-Hölder type on [a, b] and u is of bounded variation on [a, b], is proved. Applications to the approximation problem of the Riemann-Stieltjes integral in terms of Riemann-Stieltjes sums are also pointed out.

On some Chebyshev type inequalities for thecomplex integral

“…provided that f ′ , g ′ exist and are continuous on [a, b] and f ′ ∞ = sup t∈[a,b] |f ′ (t)| . The constant 1 12 cannot be improved in the general case. The Chebyshev inequality (3) also holds if f, g : [a, b] → R are assumed to be absolutely continuous and…”

A refinement of Grüss inequality for the complex integral

“…For two Lebesgue integrable functions f, g : [a, b] → C, in order to compare the integral mean of the product with the product of the integral means, we consider theČebyšev functional defined by In 1934, G. Grüss [17] showed that The constant 1 4 is best possible in (1) in the sense that it cannot be replaced by a smaller one.…”

“…The constant 1 2 is best in (5) as shown by Cerone and Dragomir in [7]. For other inequality of Grüss' type see [1]- [5], [7]- [16], [18]- [23] and [25]- [28]. In order to extend Grüss' inequality to complex integral we need the following preparations.…”

Lp–Error Bounds of Two and Three–Point Quadrature Rules For Riemann–Stieltjes Integrals

“…Motivated by Guessab-Schmeisser inequality (see [21]) which is of Ostrowski's type, Alomari in [4] and [8] presented the following approximation formula for RSintegrals:…”