Two Ostrowski type inequalities for the Stieltjes integral of monotonic functions

Abstract

Two integral inequalities of Ostrowski type for the Stieltjes integral are given. The first is for monotonic integrators and Holder continuous integrands while the second considers the dual case, that is, for monotonic integrands and Holder continuous integrators. Applications for the mid-point inequality that are useful in the numerical analysis of Stieltjes integrals are exhibited. Some connections with the generalised trapezoidal rule are also presented.

A companion of Ostrowski's inequality for complex functions defined on the unit circle and applications

“…[4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Motivated by the above facts, we consider in the present paper the problem of approximating the…”

Error bounds in approximating the Riemann–Stieltjes integral of Cn+1-class integrands and nonsmooth integrators

“…For a comprehensive recent collection of works related to Ostrowski's inequality, see the book [30], the papers [2][3][4][5][6][7][8][9][10][11]33,39,41,43].…”

APPROXIMATING THE STIELTJES INTEGRAL FOR (Φ,Φ)-Lipschitzian INTEGRATORS

“…For various bounds on the error functional D( f, u; a, b) where f and u belong to different classes of function for which the Stieltjes integral exists, see [7,[12][13][14] and the references therein. [3] Stieltjes integral for (ϕ, )-Lipschitzian integrators 75…”

“…If u is monotonic non-decreasing and f of q − K -Hölder type, then the following refinement of (1.11) also holds [7]:…”

“…If f is monotonic non-decreasing and u is of r − H -Hölder type, then [7] |θ ( f, u; a, b, The error functional T ( f, u; a, b, x) satisfies similar bounds, see [1,2,7,15] and the details are omitted.…”