Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
VolumeΒ 2012, Article IDΒ 615737, 10 pages
http://dx.doi.org/10.1155/2012/615737
Research Article

Refinements of Hermite-Hadamard Inequalities for Functions When a Power of the Absolute Value of the Second Derivative Is P-Convex

1Department of Mathematics, Lorestan University, P.O. Box 465, Khoramabad, Iran
2Department of Civil Engineering, Shahid Chamran University, P.O. Box 135, Ahvaz, Iran
3School of Engineering and Science, Victoria University, P.O. Box 14428, Melbourne, MC 8001, Australia
4School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa

Received 10 March 2012; Accepted 10 May 2012

Academic Editor: RenatΒ Zhdanov

Copyright Β© 2012 A. Barani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We extend some estimates of the right-hand side of Hermite-Hadamard-type inequalities for functions whose second derivatives absolute values are 𝑃-convex. Applications to some special means are considered.

1. Introduction

Let π‘“βˆΆπΌβ†’β„ be a convex function defined on the interval 𝐼 of real numbers and π‘Ž,π‘βˆˆπΌ with π‘Ž<𝑏. The following double inequality π‘“ξ‚€π‘Ž+𝑏2≀1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“(π‘₯)𝑑π‘₯≀𝑓(π‘Ž)+𝑓(𝑏)2(1.1) is known in the literature as the Hermite-Hadamard inequality. Both inequalities hold in the reversed direction if 𝑓 is concave. We note that Hermite-Hadamard inequality may be regarded as a refinement of the concept of convexity and it follows easily from Jensen's inequality. Note that some of the classical inequalities for means can be derived from (1.1) for appropriate particular selections of the function 𝑓. Both inequalities hold in the reversed direction if 𝑓 is concave (see [1]).

It is well known that the Hermite-Hadamard inequality plays an important role in nonlinear analysis. Over the last decade, this classical inequality has been improved and generalized in a number of ways; there have been a large number of research papers written on this subject, (see, [2–13]) and the references therein. In [13] Dragomir and Agarwal established the following results connected with the right-hand side of (1.1) as well as applied them for some elementary inequalities for real numbers and numerical integration.

Theorem 1.1. Assume that π‘Ž,π‘βˆˆβ„ with π‘Ž<𝑏 and π‘“βˆΆ[π‘Ž,𝑏]→ℝ is a differentiable function on (π‘Ž,𝑏). If |𝑓′| is convex on [π‘Ž,𝑏], then the following inequality holds: ||||𝑓(π‘Ž)+𝑓(𝑏)2βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Ž||||≀||𝑓𝑓(π‘₯)𝑑π‘₯(π‘βˆ’π‘Ž)ξ…ž||+||𝑓(π‘Ž)ξ…ž||ξ€Έ(𝑏)8.(1.2)

Theorem 1.2. Assume that π‘Ž,π‘βˆˆβ„ with π‘Ž<𝑏 and π‘“βˆΆ[π‘Ž,𝑏]→ℝ is a differentiable function on (π‘Ž,𝑏). Assume π‘βˆˆβ„ with 𝑝>1. If |𝑓′|𝑝/(π‘βˆ’1) is convex on [π‘Ž,𝑏], then the following inequality holds: ||||𝑓(π‘Ž)+𝑓(𝑏)2βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Ž||||≀𝑓(π‘₯)𝑑π‘₯π‘βˆ’π‘Ž2(𝑝+1)1/𝑝⋅||π‘“ξ…ž||(π‘Ž)𝑝/(π‘βˆ’1)+||π‘“ξ…ž||(𝑏)𝑝/(π‘βˆ’1)2ξƒ­(π‘βˆ’1)/𝑝.(1.3)

In [1] Pearce and PečariΔ‡ proved the following theorem.

Theorem 1.3. Let π‘“βˆΆπΌβ†’β„ be a differentiable function on 𝐼∘,π‘Ž,π‘βˆˆπΌβˆ˜ with π‘Ž<𝑏. If |π‘“ξ…ž|π‘ž is convex on [π‘Ž,𝑏], for π‘žβ‰₯1, then the following inequality holds: ||||𝑓(π‘Ž)+𝑓(𝑏)2βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Ž||||≀𝑓(π‘₯)𝑑π‘₯(π‘βˆ’π‘Ž)4||π‘“ξ…ž(||π‘Ž)π‘ž+||π‘“ξ…ž(||𝑏)π‘ž2ξƒͺ1/π‘ž.(1.4)

Recall that the function π‘“βˆΆ[π‘Ž,𝑏]→ℝ is said to be quasiconvex if for every π‘₯,π‘¦βˆˆπΌ we have [].𝑓(𝑑π‘₯+(1βˆ’π‘‘)𝑦)≀max{𝑓(π‘₯),𝑓(𝑦)},βˆ€π‘‘βˆˆ0,1(1.5)

The generalizations of the Theorems 1.1 and 1.2 are introduced by Ion in [14] for quasiconvex functions and are given in [6] to differentiable 𝑃-convex functions. Then, Alomari et al. in [2] improved the results in [14] and Theorem 1.3, for twice differentiable quasiconvex functions.

On the other hand, Dragomir et al. in [11] defined the following class of functions.

Definition 1.4. Let πΌβŠ†β„ be an interval. The function π‘“βˆΆπΌβ†’β„ is said to be 𝑃-convex (or belong to the class 𝑃(𝐼)) if it is nonnegative and, for all π‘₯,π‘¦βˆˆπΌ and πœ†βˆˆ[0,1], satisfies the inequality 𝑓(πœ†π‘₯+(1βˆ’πœ†)𝑦)≀𝑓(π‘₯)+𝑓(𝑦).(1.6)

Note that 𝑃(𝐼) contain all nonnegative convex and quasiconvex functions. Since then numerous articles have appeared in the literature reflecting further applications in this category, see [3, 6, 12, 15, 16] and references therein. Ozdemir and Yildiz in [15] proved the following results.

Theorem 1.5. Let π‘“βˆΆπΌβ†’β„ be a twice differentiable function on 𝐼∘ and π‘Ž,π‘βˆˆπΌβˆ˜ with π‘Ž<𝑏. If |π‘“ξ…žξ…ž| is 𝑃-convex, 0β‰€πœ†β‰€1, then the following inequality holds: ||||ξ‚€(1βˆ’πœ†)π‘“π‘Ž+𝑏2ξ‚βˆ’πœ†π‘“(π‘Ž)+𝑓(𝑏)2+1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||β‰€βŽ§βŽͺ⎨βŽͺ⎩(π‘₯)𝑑π‘₯(π‘βˆ’π‘Ž)2ξ€·248πœ†3||π‘“βˆ’3πœ†+1ξ€Έξ€½ξ…žξ…ž||+||𝑓(π‘Ž)ξ…žξ…ž||ξ€Ύ1(𝑏),0β‰€πœ†β‰€2,(π‘βˆ’π‘Ž)2ξ€½||𝑓24(3πœ†βˆ’1)ξ…žξ…ž||+||𝑓(π‘Ž)ξ…žξ…ž||ξ€Ύ,1(𝑏)2β‰€πœ†β‰€1.(1.7)

Corollary 1.6. If in Theorem 1.5 one chooses πœ†=1, one obtains ||||𝑓(π‘Ž)+𝑓(𝑏)2βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||≀(π‘₯)𝑑π‘₯(π‘βˆ’π‘Ž)2ξ€½||𝑓12ξ…žξ…ž||+||𝑓(π‘Ž)ξ…žξ…ž||ξ€Ύ.(𝑏)(1.8)

Theorem 1.7. Let π‘“βˆΆπΌβ†’β„ be a twice differentiable function on 𝐼∘ and π‘Ž,π‘βˆˆπΌβˆ˜ with π‘Ž<𝑏. If |π‘“ξ…žξ…ž|π‘ž is 𝑃-convex, 0β‰€πœ†β‰€1 and π‘žβ‰₯1, then the following inequality holds: ||||ξ‚€(1βˆ’πœ†)π‘“π‘Ž+𝑏2ξ‚βˆ’πœ†π‘“(π‘Ž)+𝑓(𝑏)2+1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||β‰€βŽ§βŽͺ⎨βŽͺ⎩(π‘₯)𝑑π‘₯(π‘βˆ’π‘Ž)2ξ€·488πœ†3||π‘“βˆ’3πœ†+1ξ€Έξ€·ξ€½ξ…žξ…ž||(π‘Ž)π‘ž+||π‘“ξ…žξ…ž||(𝑏)π‘žξ€Ύξ€Έ1/π‘ž1,0β‰€πœ†β‰€2,(π‘βˆ’π‘Ž)2||𝑓48(3πœ†βˆ’1)ξ€·ξ€½ξ…žξ…ž||(π‘Ž)π‘ž+||π‘“ξ…žξ…ž||(𝑏)π‘žξ€Ύξ€Έ1/π‘ž,12β‰€πœ†β‰€1.(1.9)

Corollary 1.8. If in Theorem 1.7 one chooses πœ†=1, one obtains ||||𝑓(π‘Ž)+𝑓(𝑏)2βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||≀(π‘₯)𝑑π‘₯(π‘βˆ’π‘Ž)2||𝑓24ξ€·ξ€½ξ…žξ…ž||(π‘Ž)π‘ž+||π‘“ξ…žξ…ž||(𝑏)π‘žξ€Ύξ€Έ1/π‘ž.(1.10)

The main purpose of this paper is to establish the refinements of results in [15]. Applications for special means are considered.

2. Main Results

In order to prove our main theorems, we need the following Lemma in [5] throughout this paper.

Lemma 2.1. Suppose that π‘“βˆΆπΌβ†’β„ is a twice differentiable function on 𝐼∘, the interior of 𝐼. Assume that π‘Ž,π‘βˆˆπΌβˆ˜, with π‘Ž<𝑏 and π‘“ξ…žξ…ž, is integrable on [π‘Ž,𝑏]. Then, the following equality holds: 𝑓(π‘Ž)+𝑓(𝑏)2βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Ž=𝑓(π‘₯)𝑑π‘₯(π‘βˆ’π‘Ž)2ξ€œ1610ξ€·1βˆ’π‘‘2ξ€Έξ‚€π‘“ξ…žξ…žξ‚€1+𝑑2π‘Ž+1βˆ’π‘‘2𝑏+π‘“ξ…žξ…žξ‚€1βˆ’π‘‘2π‘Ž+1+𝑑2𝑏𝑑𝑑.(2.1)

In the following theorem, we will propose some new upper bound for the right-hand side of (1.1) for 𝑃-convex functions, which is better than the inequality had done in [15].

Theorem 2.2. Let π‘“βˆΆπΌβ†’β„ be a twice differentiable function on 𝐼∘ such that |π‘“ξ…žξ…ž| is a 𝑃-convex function on 𝐼. Suppose that π‘Ž,π‘βˆˆπΌβˆ˜ with π‘Ž<𝑏 and π‘“ξ…žξ…žβˆˆπΏ1[π‘Ž,𝑏]. Then, the following inequality holds: ||||𝑓(π‘Ž)+𝑓(𝑏)2βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||≀(π‘₯)𝑑π‘₯(π‘βˆ’π‘Ž)2||𝑓24ξ…žξ…ž|||||𝑓(π‘Ž)+2ξ…žξ…žξ‚€π‘Ž+𝑏2|||+||π‘“ξ…žξ…ž||ξ‚„(𝑏).(2.2)

Proof. Since |π‘“ξ…žξ…ž| is a 𝑃-convex function, by using Lemma 2.1 we get ||||𝑓(π‘Ž)+𝑓(𝑏)2βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||=||||(π‘₯)𝑑π‘₯(π‘βˆ’π‘Ž)2ξ€œ1610ξ€·1βˆ’π‘‘2ξ€Έξ‚€π‘“ξ…žξ…žξ‚€1+𝑑2π‘Ž+1βˆ’π‘‘2𝑏+π‘“ξ…žξ…žξ‚€1βˆ’π‘‘2π‘Ž+1+𝑑2𝑏||||≀𝑑𝑑(π‘βˆ’π‘Ž)2ξ€œ1610||1βˆ’π‘‘2||ξ‚€|||π‘“ξ…žξ…žξ‚€1+𝑑2π‘Ž+1βˆ’π‘‘2𝑏|||+|||π‘“ξ…žξ…žξ‚€1βˆ’π‘‘2π‘Ž+1+𝑑2𝑏|||=𝑑𝑑(π‘βˆ’π‘Ž)2||𝑓24ξ…žξ…ž|||||𝑓(π‘Ž)+2ξ…žξ…žξ‚€π‘Ž+𝑏2|||+||π‘“ξ…žξ…ž||ξ‚„.(𝑏)(2.3)

An immediate consequence of Theorem 2.2 is as follows.

Corollary 2.3. Let 𝑓 be as in Theorem 2.2, if in addition

(i) π‘“ξ…žξ…ž((π‘Ž+𝑏)/2)=0, then one has
||||𝑓(π‘Ž)+𝑓(𝑏)2βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||≀(π‘₯)𝑑π‘₯(π‘βˆ’π‘Ž)2ξ€Ί||𝑓24ξ…žξ…ž||+||𝑓(𝑏)ξ…žξ…ž||ξ€»,(𝑏)(2.4)

(ii) π‘“ξ…žξ…ž(π‘Ž)=π‘“ξ…žξ…ž(𝑏)=0, then one has
||||𝑓(π‘Ž)+𝑓(𝑏)2βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||≀(π‘₯)𝑑π‘₯(π‘βˆ’π‘Ž)2|||𝑓12ξ…žξ…žξ‚€π‘Ž+𝑏2|||.(2.5)

The corresponding version for powers of the absolute value of the second derivative is incorporated in the following theorem.

Theorem 2.4. Let π‘“βˆΆπΌβ†’β„ be a differentiable function on 𝐼∘. Assume that π‘βˆˆβ„, 𝑝>1 such that |π‘“ξ…žξ…ž|𝑝/(π‘βˆ’1) is a 𝑃-convex function on 𝐼. Suppose that π‘Ž,π‘βˆˆπΌβˆ˜ with π‘Ž<𝑏 and π‘“ξ…žξ…žβˆˆπΏ1[π‘Ž,𝑏]. Then, the following inequality holds: ||||𝑓(π‘Ž)+𝑓(𝑏)2βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||≀(π‘₯)𝑑π‘₯(π‘βˆ’π‘Ž)2ξƒ©βˆš24πœ‹2ξƒͺ1/𝑝Γ(1+𝑝)ξ‚ΆΞ“(3/2+𝑝)1/𝑝||π‘“ξ…žξ…ž||(π‘Ž)π‘ž+|||π‘“ξ…žξ…žξ‚€π‘Ž+𝑏2|||π‘žξ‚Ά1/π‘ž+ξ‚΅||π‘“ξ…žξ…ž||(𝑏)π‘ž+|||π‘“ξ…žξ…žξ‚€π‘Ž+𝑏2|||π‘žξ‚Ά1/π‘žξƒ­,(2.6) where 1/𝑝+1/π‘ž=1.

Proof. By assumption, Lemma 2.1 and HΓΆlder's inequality, we have ||||𝑓(π‘Ž)+𝑓(𝑏)2βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||≀||||(π‘₯)𝑑π‘₯(π‘βˆ’π‘Ž)2ξ€œ1610ξ€·1βˆ’π‘‘2ξ€Έξ‚€π‘“ξ…žξ…žξ‚€1+𝑑2π‘Ž+1βˆ’π‘‘2𝑏+π‘“ξ…žξ…žξ‚€1βˆ’π‘‘2π‘Ž+1+𝑑2𝑏||||≀𝑑𝑑(π‘βˆ’π‘Ž)2ξ‚΅ξ€œ1610ξ€·1βˆ’π‘‘2𝑝𝑑𝑑1/π‘Γ—ξƒ¬ξ‚΅ξ€œ10|||π‘“ξ…žξ…žξ‚€1+𝑑2π‘Ž+1βˆ’π‘‘2𝑏|||π‘žξ‚Άπ‘‘π‘‘1/π‘ž+ξ‚΅ξ€œ10|||π‘“ξ…žξ…žξ‚€1βˆ’π‘‘2π‘Ž+1+𝑑2𝑏|||π‘žξ‚Άπ‘‘π‘‘1/π‘žξƒ­β‰€π‘‘π‘‘(π‘βˆ’π‘Ž)2ξƒ©βˆš24πœ‹2ξƒͺ1/𝑝Γ(1+𝑝)ξ‚ΆΞ“((3/2)+𝑝)1/𝑝×||π‘“ξ…žξ…ž||(π‘Ž)π‘ž+|||π‘“ξ…žξ…žξ‚€π‘Ž+𝑏2|||π‘žξ‚Ά1/π‘ž+ξ‚΅||π‘“ξ…žξ…ž||(𝑏)π‘ž+|||π‘“ξ…žξ…žξ‚€π‘Ž+𝑏2|||π‘žξ‚Ά1/π‘žξƒ­,(2.7) where 1/𝑝+1/π‘ž=1. We note that the Beta and Gamma functions are defined, respectively, as follows ξ€œΞ“(π‘₯)=10π‘’βˆ’π‘₯𝑑π‘₯βˆ’1π›½ξ€œπ‘‘π‘‘,π‘₯>0,(π‘₯,𝑦)=10𝑑π‘₯βˆ’1(1βˆ’π‘‘)π‘¦βˆ’1𝑑𝑑,π‘₯>0,𝑦>0(2.8) and are used to evaluate the integral ∫10(1βˆ’π‘‘2)𝑝𝑑𝑑. Indeed, by setting 𝑑2=𝑒, we get 1𝑑𝑑=2π‘’βˆ’1/2𝑑𝑒,(2.9) and using property 𝛽(π‘₯,𝑦)=Ξ“(π‘₯)Ξ“(𝑦)Ξ“(π‘₯+𝑦)(2.10) of Beta function, we obtain ξ€œ10ξ€·1βˆ’π‘‘2𝑝1𝑑𝑑=2ξ€œ10π‘’βˆ’1/2(1βˆ’π‘’)𝑝1𝑑𝑒=2𝛽12,𝑝+1=2βˆ’1Ξ“(1/2)Ξ“(1+𝑝)Ξ“(3/2+𝑝)=2βˆ’1βˆšπœ‹Ξ“(1+𝑝)Ξ“=ξƒ©βˆš(3/2+𝑝)πœ‹2ξƒͺΞ“(1+𝑝),Ξ“(3/2+𝑝)(2.11) where βˆšΞ“(1/2)=πœ‹ and the proof is completed.

The following corollary is an immediate consequence of Theorem 2.4.

Corollary 2.5. Let 𝑓 be as in Theorem 2.4, if in addition

(i) π‘“ξ…žξ…ž((π‘Ž+𝑏)/2)=0, then one has
||||𝑓(π‘Ž)+𝑓(𝑏)2βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Ž||||≀𝑓(π‘₯)𝑑π‘₯(π‘βˆ’π‘Ž)2ξƒ©βˆš24πœ‹2ξƒͺ1/𝑝Γ(1+𝑝)Ξ“ξ‚Ά(3/2+𝑝)1/𝑝||π‘“ξ…žξ…ž||+||𝑓(𝑏)ξ…žξ…ž||ξ€Έ,(𝑏)(2.12)

(ii) π‘“ξ…žξ…ž(π‘Ž)=π‘“ξ…žξ…ž(𝑏)=0, then one has
||||𝑓(π‘Ž)+𝑓(𝑏)2βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Ž||||≀𝑓(π‘₯)𝑑π‘₯(π‘βˆ’π‘Ž)2ξƒ©βˆš12πœ‹2ξƒͺ1/𝑝Γ(1+𝑝)Ξ“ξ‚Ά(3/2+𝑝)1/𝑝|||π‘“ξ…žξ…žξ‚€π‘Ž+𝑏2|||.(2.13)

Another similar result may be extended in the following theorem.

Theorem 2.6. Let π‘“βˆΆπΌβ†’β„ be a differentiable function on 𝐼∘. Assume that π‘žβ‰₯1 such that |π‘“ξ…žξ…ž|π‘ž is a 𝑃-convex function on 𝐼. Suppose that π‘Ž,π‘βˆˆπΌβˆ˜ with π‘Ž<𝑏 and π‘“ξ…žξ…žβˆˆπΏ1[π‘Ž,𝑏]. Then, the following inequality holds: ||||𝑓(π‘Ž)+𝑓(𝑏)2βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||≀(π‘₯)𝑑π‘₯(π‘βˆ’π‘Ž)2||𝑓24ξ…žξ…ž||(π‘Ž)π‘ž+|||π‘“ξ…žξ…žξ‚€π‘Ž+𝑏2|||π‘žξ‚Ά1/π‘ž+ξ‚΅||π‘“ξ…žξ…ž||(𝑏)π‘ž+|||π‘“ξ…žξ…žξ‚€π‘Ž+𝑏2|||π‘žξ‚Ά1/π‘žξƒ­.(2.14)

Proof. Suppose that π‘Ž,π‘βˆˆπΌβˆ˜. From Lemma 2.1 and using well-known power mean inequality, we get ||||𝑓(π‘Ž)+𝑓(𝑏)2βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||≀||||(π‘₯)𝑑π‘₯(π‘βˆ’π‘Ž)2ξ€œ1610ξ€·1βˆ’π‘‘2ξ€Έξ‚€π‘“ξ…žξ…žξ‚€1+𝑑2π‘Ž+1βˆ’π‘‘2𝑏+π‘“ξ…žξ…žξ‚€1βˆ’π‘‘2π‘Ž+1+𝑑2𝑏||||≀𝑑𝑑(π‘βˆ’π‘Ž)2ξ‚΅ξ€œ1610ξ€·1βˆ’π‘‘2𝑑𝑑1βˆ’1/π‘žξƒ¬ξ‚΅ξ€œ10ξ€·1βˆ’π‘‘2ξ€Έ|||π‘“ξ…žξ…žξ‚€1+𝑑2π‘Ž+1βˆ’π‘‘2𝑏|||π‘žξ‚Άπ‘‘π‘‘1/π‘ž+ξ‚΅ξ€œ10ξ€·1βˆ’π‘‘2ξ€Έ|||π‘“ξ…žξ…žξ‚€1βˆ’π‘‘2π‘Ž+1+𝑑2𝑏|||π‘žξ‚Άπ‘‘π‘‘1/π‘žξƒ­β‰€π‘‘π‘‘(π‘βˆ’π‘Ž)2ξ‚€21631βˆ’1/π‘žξƒ¬ξ‚΅23ξ‚»||π‘“ξ…žξ…ž||(π‘Ž)π‘ž+|||π‘“ξ…žξ…žξ‚€π‘Ž+𝑏2|||π‘žξ‚Όξ‚Ά1/π‘ž+ξ‚΅23ξ‚»||π‘“ξ…žξ…ž||(𝑏)π‘ž+|||π‘“ξ…žξ…žξ‚€π‘Ž+𝑏2|||π‘žξ‚Όξ‚Ά1/π‘žξƒ­=(π‘βˆ’π‘Ž)2||𝑓24ξ…žξ…ž||(π‘Ž)π‘ž+|||π‘“ξ…žξ…žξ‚€π‘Ž+𝑏2|||π‘žξ‚Ά1/π‘ž+ξ‚΅||π‘“ξ…žξ…ž||(𝑏)π‘ž+|||π‘“ξ…žξ…žξ‚€π‘Ž+𝑏2|||π‘žξ‚Ά1/π‘žξƒ­,(2.15) which completes the proof.

Corollary 2.7. Let 𝑓 be as in Theorem 2.6, if in addition(i)π‘“ξ…žξ…ž((π‘Ž+𝑏)/2)=0, then (2.4) holds,(ii)π‘“ξ…žξ…ž(π‘Ž)=π‘“ξ…žξ…ž(𝑏)=0, (2.5) holds.

3. Applications to Special Means

Now, we consider the applications of our theorems to the special means. We consider the means for arbitrary real numbers 𝛼, 𝛽 (𝛼≠𝛽). We take the following

(1) Arithmetic mean:
𝐴(𝛼,𝛽)=𝛼+𝛽2,𝛼,π›½βˆˆβ„.(3.1)

(2) Logarithmic mean:
𝐿(𝛼,𝛽)=π›Όβˆ’π›½||𝛽||||𝛽||ln|𝛼|βˆ’ln,|𝛼|β‰ ,𝛼,𝛽≠0,𝛼,π›½βˆˆβ„.(3.2)

(3) Generalized log-mean:
𝐿𝑛𝛽(𝛼,𝛽)=𝑛+1βˆ’π›Όπ‘›+1ξ‚Ή(𝑛+1)(π›½βˆ’π›Ό)1/𝑛,π‘›βˆˆβ„•,𝛼,π›½βˆˆβ„,𝛼≠𝛽.(3.3)

Now, using the results of Section 2, we give some applications for special means of real numbers.

Proposition 3.1. Let π‘Ž,π‘βˆˆβ„, π‘Ž<𝑏, and π‘›βˆˆβ„•, 𝑛β‰₯2. Then, one has ||𝐿𝑛𝑛(π‘Ž,𝑏)βˆ’π΄(π‘Žπ‘›,𝑏𝑛)||≀𝑛(π‘›βˆ’1)24(π‘βˆ’π‘Ž)2ξ‚Έ|π‘Ž|π‘›βˆ’2|||+2π‘Ž+𝑏2|||π‘›βˆ’2+||𝑏||π‘›βˆ’2ξ‚Ή.(3.4)

Proof. The assertion follows from Theorem 2.2 applied to the 𝑃-convex function 𝑓(π‘₯)=π‘₯𝑛, π‘₯βˆˆβ„.

Proposition 3.2. Let π‘Ž,π‘βˆˆβ„, π‘Ž<𝑏, and 0βˆ‰[0,1]. Then, for all 𝑝>1 one has ||πΏβˆ’1ξ€·π‘Ž(π‘Ž,𝑏)βˆ’π΄βˆ’1,π‘βˆ’1ξ€Έ||≀(π‘βˆ’π‘Ž)28ξƒ©βˆšπœ‹2ξƒͺ1/𝑝Γ(1+𝑝)ξ‚ΆΞ“((3/2)+𝑝)1/𝑝×|π‘Ž|βˆ’3π‘ž+|||π‘Ž+𝑏2|||βˆ’3π‘žξ‚Ά1/π‘ž+ξ‚΅||𝑏||βˆ’3π‘ž+|||π‘Ž+𝑏2|||βˆ’3π‘žξ‚Ά1/π‘žξƒ­,(3.5) where 1/𝑝+1/π‘ž=1.

Proof. The assertion follows from Theorem 2.4 applied to the 𝑃-convex function 𝑓(π‘₯)=1/π‘₯, π‘₯∈[π‘Ž,𝑏].

Proposition 3.3. Let π‘Ž,π‘βˆˆβ„, π‘Ž<𝑏, and π‘›βˆˆβ„•, 𝑛β‰₯2. Then, for all π‘žβ‰₯1 one has ||𝐿𝑛𝑛(π‘Ž,𝑏)βˆ’π΄(π‘Žπ‘›,𝑏𝑛)||≀𝑛(π‘›βˆ’1)24(π‘βˆ’π‘Ž)2|π‘Ž|(π‘›βˆ’2)π‘ž+|||π‘Ž+𝑏2|||(π‘›βˆ’2)π‘žξ‚Ά1/π‘ž+ξ‚΅||𝑏||(π‘›βˆ’2)π‘ž+|||π‘Ž+𝑏2|||(π‘›βˆ’2)π‘žξ‚Ά1/π‘žξƒ­.(3.6)

Proof. The assertion follows from Theorem 2.6 applied to the 𝑃-convex function 𝑓(π‘₯)=π‘₯𝑛, π‘₯βˆˆβ„.

References

  1. C. E. M. Pearce and J. Pečarić, β€œInequalities for differentiable mappings with application to special means and quadrature formula,” Applied Mathematics Letters, vol. 13, no. 2, pp. 51–55, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  2. M. Alomari, M. Darus, and S. S. Dragomir, β€œNew inequalities of Hermite-Hadamard type for functions whose second derivatives absolute values are quasi-convex,” Tamkang Journal of Mathematics, vol. 41, no. 4, pp. 353–359, 2010. View at Google Scholar Β· View at Zentralblatt MATH
  3. A. O. Akdemir and M. E. Ozdemir, β€œSome Hadamard-type inequalities for coordinated P-convex functions and Godunova-Levin functions,” in Proceedings of the AIP Conference Proceedings, vol. 1309, pp. 7–15, 2010.
  4. A. Barani, A. G. Ghazanfari, and S. S. Dragomir, β€œHermite-Hadamard inequality through prequasiinvex functions,” RGMIA Research Report Collection, vol. 14, article 48, 2011. View at Google Scholar
  5. A. Barani, S. Barani, and S. S. Dragomir, β€œRefinements of Hermite-Hadamard type inequality for functions whose second derivatives absolute values are quasiconvex,” RGMIA Research Report Collection, vol. 14, article 69, 2011. View at Google Scholar
  6. A. Barani and S. Barani, β€œHermite-Hadamard inequality for functions whose derivatives absolute values are P-convex,” Bulletin of the Australian Mathematical Society. In press. View at Publisher Β· View at Google Scholar
  7. S. S. Dragomir and C. E. M. Pearce, β€œSelected Topics on Hermite-Hadamard Inequalities and Applications,” RGMIA Monographs, Victoria University, 2000, http://ajmaa.org/RGMIA/monographs.php/.
  8. S. S. Dragomir, β€œTwo mappings in connection to Hadamard's inequalities,” Journal of Mathematical Analysis and Applications, vol. 167, no. 1, pp. 49–56, 1992. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  9. S. S. Dragomir, β€œOn Hadamard’s inequalities for convex functions,” Mathematica Balkanica, vol. 6, pp. 215–222, 1992. View at Google Scholar
  10. S. S. Dragomir, J. E. Pečarić, and J. Sándor, β€œA note on the Jensen-Hadamard inequality,” L'Analyse Numérique et la Théorie de l'Approximation, vol. 19, no. 1, pp. 29–34, 1990. View at Google Scholar Β· View at Zentralblatt MATH
  11. S. S. Dragomir, J. Pečarić, and L. E. Persson, β€œSome inequalities of Hadamard type,” Soochow Journal of Mathematics, vol. 21, no. 3, pp. 335–341, 1995. View at Google Scholar Β· View at Zentralblatt MATH
  12. S. S. Dragomir and C. E. M. Pearce, β€œQuasi-convex functions and Hadamard's inequality,” Bulletin of the Australian Mathematical Society, vol. 57, no. 3, pp. 377–385, 1998. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  13. S. S. Dragomir and R. P. Agarwal, β€œTwo inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula,” Applied Mathematics Letters. An International Journal of Rapid Publication, vol. 11, no. 5, pp. 91–95, 1998. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  14. D. A. Ion, β€œSome estimates on the Hermite-Hadamard inequality through quasi-convex functions,” Analele Universit\u a\c tii din Craiova. Seria Matematic\u a-Informatic\u a. Annals of the University of Craiova. Mathematics and Computer Science Series, vol. 34, pp. 83–88, 2007. View at Google Scholar Β· View at Zentralblatt MATH
  15. M. E. Ozdemir and C. Yildiz, β€œNew inequalities for Hermite-Hadamard and simpson type and applications,” submitted, http://arxiv.org/abs/1103.1965.
  16. K. L. Tseng, G. S. Yang, and S. S. Dragomir, β€œOn quasiconvex functions and Hadamard’s inequality,” RGMIA Research Report Collection, vol. 14, article 1, 2003. View at Google Scholar