Chinese Journal of Mathematics
Volume 2016 (2016), Article ID 4361806, 8 pages
http://dx.doi.org/10.1155/2016/4361806

Research Article

Some New Generalized Integral Inequalities for GA--Convex Functions via Hadamard Fractional Integrals

İmdat İşcan¹ and Mustafa Aydin²

¹Department of Mathematics, Faculty of Sciences and Arts, Giresun University, 28200 Giresun, Turkey
²Department of Finance-Banking and Insurance, Alucra Turan Barutçu Vocational School, Giresun University, Alucra, 28700 Giresun, Turkey

Received 22 April 2016; Revised 14 July 2016; Accepted 1 August 2016

Academic Editor: Chang-Jian Zhao

Copyright © 2016 İmdat İşcan and Mustafa Aydin. 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 prove new generalization of Hadamard, Ostrowski, and Simpson inequalities in the framework of GA--convex functions and Hadamard fractional integral.

1. Introduction

Let a real function be defined on a nonempty interval of real line . The function is said to be convex on if inequality

holds for all and

In [1], Breckner introduced -convex functions as a generalization of convex functions as follows.

Definition 1. Let be a fixed real number. A function is said to be -convex (in the second sense), or that belongs to the class , if

for all and .

Of course, -convexity means just convexity when .

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

Theorem 2. Let be a convex function defined on the interval of real numbers and with . The following double inequality holds:

Theorem 3. Let be a mapping differentiable in , the interior of I, and let with If , , then the following inequality holds:

for all

Theorem 4. Let be a four times’ continuously differentiable mapping on and Then the following inequality holds:

We will give definitions of the right and left hand side Hadamard fractional integrals which are used throughout this paper.

Definition 5. Let . The right-sided and left-sided Hadamard fractional integrals and of order with are defined by

respectively, where is the Gamma function defined by (see [2]).

In recent years, many authors have studied errors estimations for Hermite-Hadamard, Ostrowski, and Simpson inequalities; for refinements, counterparts, and generalization see [3–10].

Definition 6 (see [11, 12]). A function is said to be GA-convex (geometric-arithmetically convex) if

for all and .

Definition 7 (see [13]). For , a function is said to be GA--convex (geometric-arithmetically -convex) if

for all and .

It can be easily seen that if , GA--convexity reduces to GA-convexity.

For recent results and generalizations concerning GA-convex and GA--convex functions see [13–19].

Lemma 8 (see [20]). For and , one has

where

Let be a differentiable function on , the interior of ; in sequel of this paper we will take

where with , , , , and is Euler Gamma function.

In [21], can gave Hermite-Hadamard’s inequalities for GA-convex functions in fractional integral forms as follows.

Theorem 9. 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 [21], can obtained some new inequalities for quasi-geometrically convex functions via fractional integrals by using the following lemma.

Lemma 10. Let be a differentiable function on such that , where with . Then for all , , and one has

In this paper, we will use Lemma 10 to obtain some new inequalities on generalization of Hadamard, Ostrowski, and Simpson type inequalities for GA--convex functions via Hadamard fractional integral.

2. Generalized Integral Inequalities for Some GA--Convex Functions via Fractional Integrals

Theorem 11. Let be a differentiable function on such that , where with . If is GA--convex on in the second sense for some fixed , , , and then the following inequality for fractional integrals holds:

where

Proof. Using Lemma 10, property of the modulus, and the power-mean inequality, we have

Since is GA--convex on , we get

and by a simple computation, we have

Hence, If we use (18), (19), and (20) in (17), we obtain the desired result. This completes the proof.

Corollary 12. Under the assumptions of Theorem 11 with , inequality (15) reduces to the following inequality:

Corollary 13. Under the assumptions of Theorem 11 with and , inequality (15) reduces to the following inequality:

where

Corollary 14. Under the assumptions of Theorem 11 with , inequality (15) reduces to the following inequality:

Corollary 15. Under the assumptions of Theorem 11 with , from inequality (15), one gets the following Simpson type inequality for fractional integrals:

Corollary 16. Under the assumptions of Theorem 11 with , from inequality (15), one gets

Corollary 17. Under the assumptions of Theorem 11 with and , from inequality (15) one gets

Corollary 18. Let the assumptions of Theorem 11 hold. If for all and , then from inequality (15), one gets the following Ostrowski type inequality for fractional integrals:

for all

Theorem 19. Let be a differentiable function on such that , where with . If is GA--convex on for some fixed , , , and then the following inequality for fractional integrals holds:

where and

Proof. Using Lemma 10, property of the modulus, the Hölder inequality, and GA--convexity of , we have

where , and

Using Lemma 8, we have

Hence, if we use (32)-(33) in (31) and replacing , we obtain the desired result. This completes the proof.

Corollary 20. Under the assumptions of Theorem 19 with , inequality (29) reduces to the following inequality:

Corollary 21. Under the assumptions of Theorem 19 with and , inequality (29) reduces to the following inequality:

Corollary 22. Under the assumptions of Theorem 19 with , from inequality (29), one gets the following Simpson type inequality for fractional integrals:

Corollary 23. Under the assumptions of Theorem 19 with , from inequality (29), one gets

Corollary 24. Under the assumptions of Theorem 19 with and , from inequality (29) one gets

Corollary 25. Let the assumptions of Theorem 19 hold. If for all and , then from inequality (29), one gets the following Ostrowski type inequality for fractional integrals:

for each

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

References

W. W. Breckner, “Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer funktionen in topologischen linearen Räumen,” Publications de l'Institut Mathématique, vol. 23, pp. 13–20, 1978.

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.

M. Alomari, M. Darus, S. S. Dragomir, and P. Cerone, “Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense,” Applied Mathematics Letters, vol. 23, no. 9, pp. 1071–1076, 2010.

M. Avci, H. Kavurmaci, and M. E. Özdemir, “New inequalities of Hermite-Hadamard type via s-convex functions in the second sense with applications,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5171–5176, 2011.

S. S. Dragomir and S. Fitzpatrik, “The Hadamard's inequality for s-convex functions in the second sense,” Demonstratio Mathematica, vol. 32, no. 4, pp. 687–696, 1999.

İ. İşcan, “New estimates on generalization of some integral inequalities for s-convex functions and their applications,” International Journal of Pure and Applied Mathematics, vol. 86, no. 4, pp. 727–746, 2013.

J. Park, “Generalization of some Simpson-like type inequalities via differentiable s-convex mappings in the second sense,” International Journal of Mathematics and Mathematical Sciences, vol. 2011, Article ID 493531, 13 pages, 2011.

E. Set, “New inequalities of Ostrowski type for mapping whose derivatives are s-convex in the second sense via fractional integrals,” Computers & Mathematics with Applications, vol. 63, no. 7, pp. 1147–1154, 2012.

M. Z. Sarıkaya, E. Set, and M. E. Özdemir, “On new inequalities of Simpson's type for s-convex functions,” Computers & Mathematics with Applications, vol. 60, no. 8, pp. 2191–2199, 2010.

M. Z. Sarıkaya, E. Set, H. Yaldız, and N. Başak, “Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities,” Mathematical and Computer Modelling, vol. 57, no. 9-10, pp. 2403–2407, 2013.

C. P. Niculescu, “Convexity according to the geometric mean,” Mathematical Inequalities & Applications, vol. 3, no. 2, pp. 155–167, 2000.

C. P. Niculescu, “Convexity according to means,” Mathematical Inequalities & Applications, vol. 6, no. 4, pp. 571–579, 2003.

Y. Shuang, H.-P. Yin, and F. Qi, “Hermite-Hadamard type integral inequalities for geometric-arithmetically s-convex functions,” Analysis, vol. 33, no. 2, pp. 197–208, 2013.

J. Hua, B.-Y. Xi, and F. Qi, “Hermite-Hadamard type inequalities for geometric-arithmetically s-convex functions,” Communications of the Korean Mathematical Society, vol. 29, no. 1, pp. 51–63, 2014.

İ. İscan, “Hermite-Hadamard type inequalities for GA-s-convex functions,” Le Matematiche, vol. 69, no. 2, pp. 129–146, 2014.

M. Kunt and İ. İşcan, “On new inequalities of Hermite-Hadamard-Fejer type for GA-s-convex functions via fractional integrals,” Konuralp Jurnal of Mathematics, vol. 4, no. 1, pp. 130–139, 2016.

S. Maden, S. Turhan, and İ. İşcan, “New Hermite-Hadamard-Fejer type inequalities for GA-convex functions,” in Proceedings of the AIP Conference, vol. 1726, Antalya, Turkey, April 2016.

X.-M. Zhang, Y.-M. Chu, and X.-H. Zhang, “The Hermite-Hadamard type inequality of GA-convex functions and its application,” Journal of Inequalities and Applications, vol. 2010, Article ID 507560, 11 pages, 2010.

T.-Y. Zhang, A.-P. Ji, and F. Qi, “Some inequalities of Hermite-HADamard type for GA-convex functions with applications to means,” Le Matematiche, vol. 68, no. 1, pp. 229–239, 2013.

J. Wang, J. Deng, and M. Fečkan, “Exploring s-e-condition and applications to some Ostrowski type inequalities via Hadamard fractional integrals,” Mathematica Slovaca, vol. 64, no. 6, pp. 1381–1396, 2014.

İ. İşcan, “New general integral inequalities for quasi-geometrically convex functions via fractional integrals,” Journal of Inequalities and Applications, vol. 2013, article 491, 15 pages, 2013.