This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research Article

Approximating the Riemann-Stieltjes integral of smooth integrands and of bounded variation integrators

SS Dragomir12* and S Abelman1

Author Affiliations

1 School of Computational & Applied Mathematics, University of the Witwatersrand, Private Bag-3, Wits-2050, Johannesburg, South Africa

2 Mathematics, College of Engineering & Science, Victoria University, P.O. Box 14428, Melbourne, MC 8001, Australia

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Journal of Inequalities and Applications 2013, 2013:154  doi:10.1186/1029-242X-2013-154


The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/154


Received:20 November 2012
Accepted:20 March 2013
Published:4 April 2013

© 2013 Dragomir and Abelman; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In the present paper, we investigate the problem of approximating the Riemann-Stieltjes integral <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M1">View MathML</a> in the case when the integrand f is n-time differentiable and the derivative <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M2">View MathML</a> is either of locally bounded variation, or Lipschitzian on an interval incorporating <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3">View MathML</a>. A priory error bounds for several classes of integrators u and applications in approximating the finite Laplace-Stieltjes transform and the finite Fourier-Stieltjes sine and cosine transforms are provided as well.

MSC: 41A51, 26D15, 26D10.

Keywords:
Riemann-Stieltjes integral; Taylor’s representation; functions of bounded variation; Lipschitzian functions; integral transforms; finite Laplace-Stieltjes transform; finite Fourier-Stieltjes sine and cosine transforms

1 Introduction

The concept of Riemann-Stieltjes integral<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M4">View MathML</a>, where f is called the integrand, u is called the integrator, plays an important role in mathematics, for instance in the definition of complex integral, the representation of bounded linear functionals on the Banach space of all continuous functions on an interval <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3">View MathML</a>, in the spectral representation of selfadjoint operators on complex Hilbert spaces and other classes of operators such as the unitary operators, etc.

However, the numerical analysis of this integral is quite poor as pointed out by the seminal paper due to Michael Tortorella from 1990 [1]. Earlier results in this direction, however, were provided by Dubuc and Todor in their 1984 and 1987 papers [2,3] and [4], respectively. For recent results concerning the approximation of the Riemann-Stieltjes integral, see the work of Diethelm [5], Liu [6], Mercer [7], Munteanu [8], Mozyrska et al.[9] and the references therein. For other recent results obtained in the same direction by the first author and his colleagues from RGMIA, see [10-16] and [17]. A comprehensive list of preprints related to this subject may be found at http://rgmia.org webcite.

In order to approximate the Riemann-Stieltjes integral <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M6">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M7">View MathML</a> are functions for which the above integral exists, Dragomir established in [18] the following integral identity:

(1.1)

provided that the involved integrals exist. In the particular case when <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M9">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M10">View MathML</a>, the above identity reduces to the celebrated Montgomery identity (see [[19], p.565]) that has been extensively used by many authors in obtaining various inequalities of Ostrowski type. For a comprehensive recent collection of works related to Ostrowski’s inequality, see the book [20], the papers [10-12,21-32] and [33]. For other results concerning error bounds of quadrature rules related to midpoint and trapezoid rules, see [34-45] and the references therein.

Motivated by the recent results from [18,46,47] (see also [11,27] and [13]) in the present paper we investigate the problem of approximating the Riemann-Stieltjes integral <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M11">View MathML</a> in the case when the integrand f is n-times differentiable and the derivative <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M2">View MathML</a> is either of locally bounded variation, or Lipschitzian on an interval incorporating <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3">View MathML</a>. A priori error bounds for several classes of integrators u and applications in approximating the finite Laplace-Stieltjes transform and the finite Fourier-Stieltjes sine and cosine transforms are provided as well.

2 Some representation results

In this section, we establish some representation results for the Riemann-Stieltjes integral when the integrand is n-times differentiable and the integrator is of locally bounded variation. Several particular cases of interest are considered as well.

Theorem 1Assume that the function<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M14">View MathML</a>isn-times differentiable on the interiorof the intervalI (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M16">View MathML</a>) and thenth derivative<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M2">View MathML</a>is of locally bounded variation on. Ifwith<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M20">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M22">View MathML</a>is of bounded variation on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3">View MathML</a>, then the Riemann-Stieltjes integral<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M1">View MathML</a>exists, we have the identity

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M25">View MathML</a>

(2.1)

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M26">View MathML</a>

(2.2)

and the remainder<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M27">View MathML</a>can be represented as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M28">View MathML</a>

(2.3)

Both integrals in (2.3) are taken in the Riemann-Stieltjes sense.

Proof

Under the assumption of the theorem, we utilize the following Taylor’s representation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M29">View MathML</a>

(2.4)

that holds for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M31">View MathML</a>. The integral in (2.4) is taken in the Riemann-Stieltjes sense.

We can prove this equality by induction.

Indeed, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M32">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M33">View MathML</a>

that holds for any function of locally bounded variation on .

Now, assume that (2.4) is true for an <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M31">View MathML</a> and let us prove that it holds for ‘<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M36">View MathML</a>’, namely

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M37">View MathML</a>

(2.5)

provided that the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M14">View MathML</a> is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M39">View MathML</a>-times differentiable on the interior of the interval I and the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M41">View MathML</a>-th derivative <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M42">View MathML</a> is of locally bounded variation on .

Utilizing the integration by parts formula for the Riemann-Stieltjes integral and the reduction of the Riemann-Stieltjes integral to a Riemann integral (see, for instance, [48]) we have:

(2.6)

From (2.4), we have that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M45">View MathML</a>

which inserted in the last part of (2.6) provides the equality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M46">View MathML</a>

(2.7)

We observe that, by division with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M47">View MathML</a>, the equality (2.7) becomes the desired representation (2.5).

Further on, from the identity (2.4) we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M48">View MathML</a>

(2.8)

Utilizing the integration by parts formula, we have for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M49">View MathML</a> that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M50">View MathML</a>

(2.9)

For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M51">View MathML</a>, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M52">View MathML</a>.

Therefore, by (2.9) we get

(2.10)

and by (2.8) the representation (2.1) is thus obtained.

This completes the proof. □

Remark 1 Assume that the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M14">View MathML</a> is n-times differentiable on the interior of the interval I (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M16">View MathML</a>) and the nth derivative <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M2">View MathML</a> is of locally bounded variation on . If with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M20">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M22">View MathML</a> is of bounded variation on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3">View MathML</a>, then, by choosing <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M63">View MathML</a> in the formulae above we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M64">View MathML</a>

(2.11)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M65">View MathML</a>

(2.12)

This give the representation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M66">View MathML</a>

(2.13)

Now, if we choose <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M67">View MathML</a>, then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M68">View MathML</a>

(2.14)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M69">View MathML</a>

(2.15)

which provide the representation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M70">View MathML</a>

(2.16)

Finally, if we choose <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M71">View MathML</a>, then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M72">View MathML</a>

(2.17)

and the remainder

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M73">View MathML</a>

(2.18)

Making use of (2.1) we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M74">View MathML</a>

(2.19)

3 Error bounds

In order to provide sharp error bounds in the approximation rules outlined above, we need the following well-known lemma concerning sharp estimates for the Riemann-Stieltjes integral for various pairs of integrands and integrators (see, for instance, [48]).

Lemma 1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M75">View MathML</a>two bounded functions on the compact interval<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3">View MathML</a>.

(i) Ifpis continuous andvis of bounded variation, then the Riemann-Stieltjes integral<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M77">View MathML</a>exists and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M78">View MathML</a>

(3.1)

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M79">View MathML</a>denotes the total variation ofvon the interval<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3">View MathML</a>.

(ii) Ifpis Riemann integrable andvis Lipschitzian with the constant<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M81">View MathML</a>, i.e.,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M82">View MathML</a>

then the Riemann-Stieltjes integral<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M83">View MathML</a>exists and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M84">View MathML</a>

(3.2)

All the above inequalities are sharp in the sense that there are examples of functions for which each equality case is realized.

Utilizing this result concerning bounds for the Riemann-Stieltjes integral, we can provide the following error bounds in approximating the integral <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M85">View MathML</a>.

Theorem 2Assume that the function<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M14">View MathML</a>isn-times differentiable on the interiorof the intervalI (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M16">View MathML</a>) and thenth derivative<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M2">View MathML</a>is of locally bounded variation on. Ifwith<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M20">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M22">View MathML</a>is of bounded variation on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3">View MathML</a>, then we have the representation (2.1), where the approximation term<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M96">View MathML</a>is given by (2.2) and the remainder<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M97">View MathML</a>satisfies the inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M98">View MathML</a>

(3.3)

for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21">View MathML</a>.

If thenth derivative<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M2">View MathML</a>is Lipschitzian with the constant<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M101">View MathML</a>on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3">View MathML</a>, then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M103">View MathML</a>

(3.4)

for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21">View MathML</a>.

Proof

Utilizing the property (i) from Lemma 1, we have successively

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M105">View MathML</a>

(3.5)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21">View MathML</a>.

For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M107">View MathML</a>, denote

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M108">View MathML</a>

(3.6)

By the property (i) from Lemma 1 applied for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M109">View MathML</a> we have for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M110">View MathML</a> that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M111">View MathML</a>

and for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M112">View MathML</a> that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M113">View MathML</a>

Therefore,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M114">View MathML</a>

(3.7)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21">View MathML</a>.

Utilizing (3.5) and (3.7), we deduce the desired inequality (3.3).

By the property (ii) from Lemma 1 applied for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M116">View MathML</a>, we have that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M117">View MathML</a>

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M107">View MathML</a>, which produces the bound

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M119">View MathML</a>

(3.8)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21">View MathML</a>.

Utilizing (3.5) and (3.8), we deduce the desired inequality (3.4). □

The best error bounds we can get from Theorem 2 are as follows.

Corollary 1Under the assumptions of Theorem 2 we have the representation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M121">View MathML</a>

(3.9)

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M122">View MathML</a>is defined in (2.14) and the error<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M123">View MathML</a>satisfies the bound

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M124">View MathML</a>

(3.10)

Moreover, if thenth derivative<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M2">View MathML</a>is Lipschitzian with the constant<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M101">View MathML</a>on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3">View MathML</a>, then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M128">View MathML</a>

(3.11)

The case of Lipschitzian integrators may be of interest as well and will be considered in the following.

Theorem 3Assume that the function<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M14">View MathML</a>isn-times differentiable on the interiorof the intervalI (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M16">View MathML</a>) and thenth derivative<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M2">View MathML</a>is of locally bounded variation on. Ifwith<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M20">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M22">View MathML</a>is Lipschitzian on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3">View MathML</a>with the constant<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M139">View MathML</a>then we have the representation (2.1), where the approximation term<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M140">View MathML</a>is given by (2.2) and the remainder<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M97">View MathML</a>satisfies the inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M142">View MathML</a>

(3.12)

for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21">View MathML</a>.

If thenth derivative<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M2">View MathML</a>is Lipschitzian with the constant<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M101">View MathML</a>on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3">View MathML</a>, then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M147">View MathML</a>

(3.13)

for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21">View MathML</a>.

Proof

Utilizing the property (ii) from Lemma 1, we have successively

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M149">View MathML</a>

(3.14)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21">View MathML</a>, where as above <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M151">View MathML</a>, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M107">View MathML</a>.

By the property (i) from Lemma 1 applied for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M109">View MathML</a>, we have for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M110">View MathML</a> that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M155">View MathML</a>

and for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M112">View MathML</a> that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M157">View MathML</a>

which gives that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M158">View MathML</a>

for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M107">View MathML</a>.

This implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M160">View MathML</a>

(3.15)

for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21">View MathML</a>.

Making use of (3.14) and (3.15) we deduce the desired inequality (3.12).

By the property (ii) from Lemma 1 applied for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M116">View MathML</a> we have that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M163">View MathML</a>

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M107">View MathML</a>, which produces the bound

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M165">View MathML</a>

(3.16)

for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21">View MathML</a>.

Utilizing (3.14) and (3.16), we deduce the desired inequality (3.13). □

The following particular case provides the best error bounds.

Corollary 2Under the assumptions of Theorem 3, we have the representation (3.9), where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M122">View MathML</a>is defined in (2.14) and the error<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M123">View MathML</a>satisfies the bound

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M169">View MathML</a>

(3.17)

Moreover, if thenth derivative<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M2">View MathML</a>is Lipschitzian with the constant<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M101">View MathML</a>on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3">View MathML</a>, then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M173">View MathML</a>

(3.18)

4 Applications

1. We consider the following finite Laplace-Stieltjes transform defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M174">View MathML</a>

(4.1)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M175">View MathML</a> are real numbers with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M20">View MathML</a>, s is a complex number and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M177">View MathML</a> is a function of bounded variation.

It is important to notice that, in the particular case <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M178">View MathML</a>, (4.1) becomes the finite Laplace transform which has various applications in other fields of Mathematics; see, for instance, [25,26,49-51] and [52] and the references therein. Therefore, any approximation of the more general finite Laplace-Stieltjes transform can be used for the particular case of finite Laplace transform.

Since the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M179">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M180">View MathML</a> is continuous for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M181">View MathML</a>, the transform (4.1) is well defined for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M181">View MathML</a>.

We observe that the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M183">View MathML</a> has derivatives of all orders and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M184">View MathML</a>

(4.2)

We also observe that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M185">View MathML</a>

To simplify the notations, we denote by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M186">View MathML</a>

(4.3)

On utilizing Theorem 1, we have the representation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M187">View MathML</a>

(4.4)

where

(4.5)

and the remainder <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M189">View MathML</a> can be represented as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M190">View MathML</a>

(4.6)

Here, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M181">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21">View MathML</a>.

Since g is of bounded variation on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3">View MathML</a> and the derivative <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M194">View MathML</a> is Lipschitzian with the constant

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M195">View MathML</a>

then by Theorem 2 we have the bound

(4.7)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M181">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21">View MathML</a>.

As above, the best approximation we can get from (4.4) is for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M67">View MathML</a>, namely, we have the representation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M200">View MathML</a>

(4.8)

where

(4.9)

and the remainder <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M202">View MathML</a> can be represented as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M203">View MathML</a>

(4.10)

The error <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M204">View MathML</a> satisfies the bound

(4.11)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M181">View MathML</a>.

Now, if we restrict the function g to belong to the class of Lipschitzian functions with the constant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M139">View MathML</a> on the interval <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M208">View MathML</a>, then the error in the representation (4.4) will satisfy the bound

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M209">View MathML</a>

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M181">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21">View MathML</a>.

Finally, the error <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M202">View MathML</a> from the representation (4.8) satisfies the inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M213">View MathML</a>

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M181">View MathML</a>.

2. We consider now the finite Fourier-Stieltjes sine and cosine transforms defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M215">View MathML</a>

(4.12)

where a, b are real numbers with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M20">View MathML</a>, u is a real number and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M217">View MathML</a> is a function of bounded variation.

Since the functions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M218">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M219">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M220">View MathML</a> are continuous for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M221">View MathML</a>, the transforms (4.12) are well defined for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M221">View MathML</a>.

Utilizing the well-known formulae for the nth derivatives of sine and cosine functions, namely,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M223">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M224">View MathML</a>

then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M225">View MathML</a>

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M221">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M227">View MathML</a>.

We observe that, in general, we have the bounds

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M228">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M229">View MathML</a>

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M221">View MathML</a>, the closed interval <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M31">View MathML</a>.

On utilizing Theorem 1, we have the representation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M233">View MathML</a>

(4.13)

where

(4.14)

and the remainder <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M235">View MathML</a> can be represented as

(4.15)

Since g is of bounded variation on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3">View MathML</a> and the derivative <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M194">View MathML</a> is Lipschitzian with the constant

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M239">View MathML</a>

then by Theorem 2 we have the bound

(4.16)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M221">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21">View MathML</a>.

As above, the best approximation we can get from (4.4) is for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M67">View MathML</a>, namely, we have the representation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M244">View MathML</a>

(4.17)

where

(4.18)

and the remainder <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M246">View MathML</a> can be represented as

(4.19)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M221">View MathML</a>.

Here, the error satisfies the bound

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M249">View MathML</a>

(4.20)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M221">View MathML</a>.

Now, if we restrict the function g to belong to the class of Lipschitzian functions with the constant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M139">View MathML</a> on the interval <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M208">View MathML</a>, then the error in the representation (4.17) will satisfy the bound:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M253">View MathML</a>

(4.21)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M221">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M21">View MathML</a>.

Finally, the error from the representation (4.17) satisfies the inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M256">View MathML</a>

(4.22)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M221">View MathML</a>.

Similar results may be stated for the finite Fourier-Stieltjes cosine transform, however the details are left to the interested reader.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors SSD and SA have contributed equally in all stages of writing the paper.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors would like to thank the anonymous referees for the valuable suggestions that have been incorporated in the final version of the paper.

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