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Sharp error bounds in approximating the Riemann-Stieltjes integral by a generalised trapezoid formula and applications

Pietro Cerone1* and Sever S Dragomir23

Author Affiliations

1 Department of Mathematics and Statistics, La Trobe University, Melbourne, VIC, 3086, Australia

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

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

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Journal of Inequalities and Applications 2013, 2013:53  doi:10.1186/1029-242X-2013-53


The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/53


Received:18 July 2012
Accepted:22 January 2013
Published:18 February 2013

© 2013 Cerone and Dragomir; 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

Sharp error bounds in approximating the Riemann-Stieltjes integral <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M1">View MathML</a> with the generalised trapezoid formula <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M2">View MathML</a> are given for various pairs <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M3">View MathML</a> of functions. Applications for weighted integrals are also provided.

MSC: 26D15, 26D10, 41A55.

Keywords:
Riemann-Stieltjes integral; trapezoid rule; integral inequalities; weighted integrals

1 Introduction

In [1], in order to approximate the Riemann-Stieltjes integral <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M4">View MathML</a> by the generalised trapezoid formula

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M5">View MathML</a>

(1.1)

the authors considered the error functional

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M6">View MathML</a>

(1.2)

and proved that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M7">View MathML</a>

(1.3)

provided that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M8">View MathML</a> is of bounded variation on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9">View MathML</a> and u is of r-H-Hölder type, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M10">View MathML</a> satisfies the condition <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M11">View MathML</a> for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M12">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M13">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M14">View MathML</a> are given.

The dual case, namely, when f is of q-K-Hölder type and u is of bounded variation, has been considered by the authors in [2] in which they obtained the bound:

(1.4)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M16">View MathML</a>.

The case where f is monotonic and u is of r-H-Hölder type, which provides a refinement for (1.3), and respectively the case where u is monotonic and f of q-K-Hölder type were considered by Cheung and Dragomir in [3], while the case where one function was of Hölder type and the other was Lipschitzian was considered in [4]. For other recent results in estimating the error <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M17">View MathML</a> for absolutely continuous integrands f and integrators u of bounded variation, see [5] and [6].

The main aim of the present paper is to investigate the error bounds in approximating the Stieltjes integral by a different generalised trapezoid rule than the one from (1.1) in which the value <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M18">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M16">View MathML</a> is replaced with the integral mean <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M20">View MathML</a>. Applications in approximating the weighted integrals <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M21">View MathML</a> are also provided.

2 Representation results

We consider the following error functional <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M22">View MathML</a> in approximating the Riemann-Stieltjes integral <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M23">View MathML</a> by the generalised trapezoid formula:

(2.1)

If we consider the associated functions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M25">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M26">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M27">View MathML</a> defined by

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M29">View MathML</a>

then we observe that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M30">View MathML</a>

(2.2)

The following representation result can be stated.

Theorem 1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M31">View MathML</a>be bounded on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9">View MathML</a>and such that the Riemann-Stieltjes integral<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M4">View MathML</a>and the Riemann integral<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M34">View MathML</a>exist. Then we have the identities

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M35">View MathML</a>

(2.3)
where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M36">View MathML</a>

(2.4)

Proof

Integrating the Riemann-Stieltjes integral by parts, we have

and the first equality in (2.3) is proved.

The second and third identity is obvious by the relation (2.2).

For the last equality, we use the fact that for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M38">View MathML</a> bounded functions for which the Riemann-Stieltjes integral <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M39">View MathML</a> and the Riemann integral <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M40">View MathML</a> exist, we have the representation (see, for instance, [7])

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M41">View MathML</a>

(2.5)

The proof is now complete. □

In the case where u is an integral, the following identity can be stated.

Corollary 1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M42">View MathML</a>be continuous on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M8">View MathML</a>be Riemann integrable. Then we have the identity

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M45">View MathML</a>

(2.6)

Proof Since p and h are continuous, the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M46">View MathML</a> is differentiable and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M47">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M48">View MathML</a>.

Integrating by parts, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M49">View MathML</a>

Since

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M50">View MathML</a>

then, by the definition of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M51">View MathML</a> in (2.1), we deduce the first part of (2.6).

The second part of (2.6) follows by (2.3). □

Remark 1 In the particular case <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M52">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M53">View MathML</a>, we have the equality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M54">View MathML</a>

(2.7)

3 Some inequalities for f-convex

The following result concerning the nonnegativity of the error functional <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M55">View MathML</a> can be stated.

Theorem 2Ifuis monotonic nonincreasing and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M56">View MathML</a>is such that the Riemann-Stieltjes integral<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M1">View MathML</a>exists and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M58">View MathML</a>

(3.1)

then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M59">View MathML</a>or, equivalently,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M60">View MathML</a>

(3.2)

A sufficient condition for (3.1) to hold is thatfis convex on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9">View MathML</a>.

Proof The condition (3.1) is equivalent with the fact that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M62">View MathML</a> for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M48">View MathML</a> and then, by the equality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M64">View MathML</a>

we deduce that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M59">View MathML</a>.

If f is convex, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M66">View MathML</a>

which shows that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M67">View MathML</a>, namely, the condition (3.1) is satisfied. □

Corollary 2Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M42">View MathML</a>be continuous on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M8">View MathML</a>be Riemann integrable. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M71">View MathML</a>for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M72">View MathML</a>andfsatisfies (3.1) or, sufficiently, fis convex on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9">View MathML</a>, then

(3.3)

We are now able to provide some new results.

Theorem 3Assume thatpandhare continuous and synchronous (asynchronous) on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M75">View MathML</a>, i.e.,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M76">View MathML</a>

(3.4)

Iffsatisfies (3.1) and is Riemann integrable on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M77">View MathML</a> (or sufficiently, fis convex on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9">View MathML</a>), then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M79">View MathML</a>

(3.5)
where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M80">View MathML</a>

(3.6)

Proof

We use the Čebyšev inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M81">View MathML</a>

(3.7)

which holds for synchronous (asynchronous) functions p, h and nonnegative α for which the involved integrals exist.

Now, on applying the Čebyšev inequality (3.7) for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M82">View MathML</a> and utilising the representation result (2.6), we deduce the desired inequality (3.5). □

We also have the following theorem.

Theorem 4Assume that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M8">View MathML</a>is Riemann integrable and satisfies (3.1) (or sufficiently, fis concave on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9">View MathML</a>). Then, for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M85">View MathML</a>continuous, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M86">View MathML</a>

(3.8)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M87">View MathML</a>

(3.9)

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M88">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M89">View MathML</a>. In particular, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M90">View MathML</a>

(3.10)

Proof

Observe that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M91">View MathML</a>

and the inequality (3.8) is proved.

Further, by the Hölder inequality, we also have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M92">View MathML</a>

for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M88">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M89">View MathML</a>, and the theorem is proved. □

Remark 2 The above result can be useful for providing some error estimates in approximating the weighted integral <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M95">View MathML</a> by the generalised trapezoid rule

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M96">View MathML</a>

as follows:

(3.11)

provided f satisfies (3.1) and is Riemann integrable (or sufficiently, convex on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9">View MathML</a>), which is continuous on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M99">View MathML</a>.

If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M100">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M72">View MathML</a>, then for some f, we also have

(3.12)

with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M88">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M89">View MathML</a>.

Finally, we can state the following Jensen type inequality for the error functional <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M105">View MathML</a>.

Theorem 5Assume<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M8">View MathML</a>is Riemann integrable and satisfies (3.1) (or sufficiently, fis convex on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9">View MathML</a>), while<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M108">View MathML</a>is continuous. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M109">View MathML</a>is convex (concave), then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M110">View MathML</a>

(3.13)

Proof

By the use of Jensen’s integral inequality, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M111">View MathML</a>

(3.14)

Since, by the identity (2.6), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M112">View MathML</a>

then (3.14) is equivalent with the desired result (3.13). □

4 Sharp bounds via Grüss type inequalities

Due to the identity (2.3), in which the error bound <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M113">View MathML</a> can be represented as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M114">View MathML</a>, where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M115">View MathML</a>

is a Grüss type functional introduced in [8], any sharp bound for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M114">View MathML</a> will be a sharp bound for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M22">View MathML</a>.

We can state the following result.

Theorem 6Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M31">View MathML</a>be bounded functions on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9">View MathML</a>.

(i) If there exist constantsn, Nsuch that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M120">View MathML</a>for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M72">View MathML</a>, uis Riemann integrable andfisK-Lipschitzian (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M122">View MathML</a>), then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M123">View MathML</a>

(4.1)

The constant<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M124">View MathML</a>is best possible in (4.1).

(ii) Iffis of bounded variation anduisS-Lipschitzian (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M125">View MathML</a>), then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M126">View MathML</a>

(4.2)

The constant<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M124">View MathML</a>is best possible in (4.2)

(iii) Iffis monotonic nondecreasing anduisS-Lipschitzian, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M128">View MathML</a>

(4.3)

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M129">View MathML</a>

The constant<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M124">View MathML</a>is best possible in both inequalities.

(iv) Iffis monotonic nondecreasing anduis of bounded variation and such that the Riemann-Stieltjes integral<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M131">View MathML</a>exists, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M132">View MathML</a>

(4.4)

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M133">View MathML</a>

The inequality (4.4) is sharp.

(v) Iffis continuous and convex on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9">View MathML</a>anduis of bounded variation on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M136">View MathML</a>

(4.5)

The constant<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M137">View MathML</a>is sharp (if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M138">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M139">View MathML</a>are finite).

(vi) If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M8">View MathML</a>is continuous and convex on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9">View MathML</a>anduis monotonic nondecreasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M143">View MathML</a>

(4.6)

The constants 2 and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M124">View MathML</a>are best possible in (4.6) (if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M138">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M139">View MathML</a>are finite).

Proof The inequality (4.1) follows from the inequality (2.5) in [8] applied to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M114">View MathML</a>, while (4.2) comes from (1.3) of [9]. The inequalities (4.3) and (4.4) follow from [7], while (4.5) and (4.6) are valid via the inequalities (2.8) and (2.1) from [10] applied to the functional <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M148">View MathML</a>. The details are omitted. □

If we consider the error functional in approximating the weighted integral <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M21">View MathML</a> by the generalised trapezoid formula,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M150">View MathML</a>

namely (see also (2.7)),

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M151">View MathML</a>

(4.7)

then the following corollary provides various sharp bounds for the absolute value of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M152">View MathML</a>.

Corollary 3Assume thatfanduare Riemann integrable on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M153">View MathML</a>.

(i) If there exist constantsγ, Γ such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M154">View MathML</a>for each<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M53">View MathML</a>, andfisK-Lipschitzian on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M157">View MathML</a>

(4.8)

The constant<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M124">View MathML</a>is best possible in (4.8).

(ii) Iffis of bounded variation and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M159">View MathML</a>for each<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M72">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M161">View MathML</a>

(4.9)

The constant<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M124">View MathML</a>is best possible in (4.9).

(iii) Iffis monotonic nondecreasing and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M163">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M72">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M165">View MathML</a>

(4.10)

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M166">View MathML</a>is defined in Theorem 6. The constant<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M124">View MathML</a>is sharp in both inequalities.

(iv) Iffis monotonic nondecreasing and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M168">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M169">View MathML</a>

(4.11)

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M170">View MathML</a>is defined in Theorem 6. The inequality (4.11) is sharp.

(v) Iffis continuous and convex on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M172">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M173">View MathML</a>

(4.12)

The constant<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M137">View MathML</a>is sharp (if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M138">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M139">View MathML</a>are finite).

(vi) If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M8">View MathML</a>is continuous and convex on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M9">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M179">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M72">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M181">View MathML</a>

(4.13)

The first inequality in (4.13) is sharp (if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M182">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M139">View MathML</a>are finite).

Proof We only prove the first inequality in (4.13).

Utilising the inequality (4.6) for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M184">View MathML</a>, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/53/mathml/M185">View MathML</a>

(4.14)

However, on integrating by parts, we have

The rest of the inequality is obvious. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

PC and SSD have contributed to all parts of the article. Both authors read and approved the final manuscript.

Acknowledgements

Most of the work for this article was undertaken while the first author was at Victoria University, Melbourne Australia.

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