Four Points Inequalities for Riemann-Stieltjes Integral of Lipschitzian Integrand and Integrator of r-H-Hölder Type with Applications

In this paper we investigate some simple error bounds in approximating the Riemann-Stieltjes integral ∫ₐᵇ f(t) du(t) by the use of a four-point formula [u((a + b)/2) − u((1 − λ)x + λa)]f(x) + [u((1 − λ)x + λa) − u(a)]f(a) + [u((1 − α)(a + b − x) + αb) − u((a + b)/2)]f(a + b − x) + [u(b) − u((1 − α)(a + b − x) + αb)]f(b), where λ, α ∈ [0, 1], x ∈ [a, (a + b)/2], and assuming that the function f is L-Lipschitzian and u is r-H-Hölder type on [a, b]. The important case of weighted integrals is considered, compounding quadrature rules are provided and applications for approximation of Fourier transforms on finite intervals are also given.