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International Journal of Analysis
Volume 2014, Article ID 353924, 8 pages
http://dx.doi.org/10.1155/2014/353924
Research Article

New General Integral Inequalities for Lipschitzian Functions via Hadamard Fractional Integrals

Department of Mathematics, Faculty of Arts and Sciences, Giresun University, 28100 Giresun, Turkey

Received 13 January 2014; Revised 18 March 2014; Accepted 1 April 2014; Published 22 April 2014

Academic Editor: Julien Salomon

Copyright © 2014 İmdat İşcan. 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

The author obtains new estimates on generalization of Hadamard, Ostrowski, and Simpson type inequalities for Lipschitzian functions via Hadamard fractional integrals. Some applications to special means of positive real numbers are also given.

1. Introduction

Let real function be defined on some nonempty interval of real line . The function is said to be convex on if inequality holds for all and .

Following are inequalities which are well known in the literature as Hermite-Hadamard inequality, Ostrowski inequality, and Simpson inequality, respectively.

Theorem 1. Let be a convex function defined on the interval of real numbers and with . The following double inequality holds:

Theorem 2. Let be a mapping differentiable in , the interior of I, and let with . If , ; then the following inequality holds: for all .

Theorem 3. Let be four times continuously differentiable mapping on and . Then the following inequality holds:

In recent years, many authors have studied errors estimations for Hermite-Hadamard, Ostrowski, and Simpson inequalities; for refinements, counterparts, and generalization, see [19] and references therein.

The following definitions are well known in the literature.

Definition 4. A function is called an -Lipschitzian function on the interval of real numbers with if for all .

For some recent results connected with Hermite-Hadamard type integral inequalities for Lipschitzian functions, see [1013].

Definition 5 (see [14, 15]). A function is said to be GA-convex (geometric arithmetically convex) if for all and .

We will now give definitions of the right-sided and left-sided Hadamard fractional integrals which are used throughout this paper.

Definition 6. Let . The right-sided and left-sided Hadamard fractional integrals and of oder with are defined by respectively, where is Gamma function defined by (see [16]).

In [17], Iscan established Hermite-Hadamard’s inequalities for GA-convex functions in Hadamard fractional integral forms as follows.

Theorem 7. Let be a function such that , where with . If is a GA-convex function on , then the following inequalities for fractional integrals hold: with .

In the inequality (8), if we take , then we have the following inequality:

Morever, in [17], Iscan obtained a generalization of Hadamard, Ostrowski, and Simpson type inequalities for quasi-geometrically convex functions via Hadamard fractional integrals as related to the inequality (8).

In this paper, the author obtains new general inequalities for Lipschitzian functions via Hadamard fractional integrals as related to the inequality (8).

2. Main Results

Let be a -Lipschitzian function on ; throughout this section, we will take where with , , , , and is Euler Gamma function.

Theorem 8. Let be a -Lipschitzian function on and with . Then for all , , and , we have the following inequality for Hadamard fractional integrals:

Proof. Using the hypothesis of , we have the following inequality:

Corollary 9. In Theorem 8, If we take , then we get In this inequality,(i)if we take , then (ii)If we take , then (iii)If we take and , then

Corollary 10. In Theorem 8, if we take , then we get In this inequality, if we take , then
Specially if we take in this inequality, then we have

Corollary 11. In Theorem 8,(1)if we take and , then Specially, if we take in this inequality, then we have (2)If we take and , then Specially, if we take in this inequality, then we have

Corollary 12. In Theorem 8, If we take , then  Specially, if we take in this inequality, then we have We note that if we take , , , and in inequality (25) we obtain inequalities (16), (19), (21), and (23), respectively.

Let be an -Lipschitzian function. In the next theorem, let , , and define , as follows.(1)If , then (2)If , then (3)If , then

Now, we shall give another result for Lipschitzian functions as follows.

Theorem 13. Let and function be defined as above. Then we have the following inequality for Hadamard fractional integrals:

Proof. Using the hypothesis of , we have the following inequality: Now, using simple calculations, we obtain the following identities and .(1)If , then (2)If , then (3)If , then Using inequality (30) and the above identities and , we derive inequality (29). This completes the proof.

Under the assumptions of Theorem 13, we have the following corollaries and remarks.

Corollary 14. In Theorem 13, if we take , then inequality (29) reduces the following inequality:

Corollary 15. In Theorem 13, let , , and . Then, we have the inequality as follows.(i)If , then (ii)If , then (iii)If , then

Corollary 16. In Theorem 13, if we take , then we have the inequality

Remark 17. In inequality (39), if we choose , , then we get inequality (15).

Corollary 18. In inequality (35), if we take , then we have the following weighted Hadamard-type inequalities for Lipschitzian functions via Hadamard fractional integrals:

Remark 19. In inequality (40), if we choose , , then we get inequality (18).

3. Application to Special Means

Let us recall the following special means of two positive numbers , with .(1)The arithmetic mean: (2)The geometric mean: (3)The harmonic mean: (4)The logarithmic mean: (5)The identric mean:

To prove the results of this section, we need the following lemma.

Lemma 20 (see [12]). Let be differentiable with . Then is an -Lipschitzian function on , where .

Proposition 21. For , and , we have

Proof. The proof follows by inequality (25) applied for the Lipschitzian function on .

Remark 22. Let and in inequality (46). Then, using inequality (9), we have the following inequalities, respectively,

Proposition 23. For and , we have

Proof. The proof follows by inequality (25) applied for the Lipschitzian function on .

Remark 24. Let and in inequality (49). Then, using inequality (9), we have the following inequalities, respectively,

Proposition 25. For , , and , we have

Proof. The proof follows by inequality (25) applied for the Lipschitzian function on .

Remark 26. Let and in inequality (51). Then, using inequality (9), we have the following inequalities, respectively,

Proposition 27. For and , we have

Proof. The proof follows by inequality (25) applied for the Lipschitzian function on .

Proposition 28. For and , we have

Proof. The proof follows by inequality (25) applied for Lipschitzian function on .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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