**Turkish Journal of Analysis and Number Theory**

##
Fractional Integral Inequalities via *s*-Convex Functions

**Çetin Yildiz**^{1,}, **M. Emіn Özdemіr**^{2}, **Havva Kavurmaci Önalan**^{3}

^{1}Atatürk University, K.K. Education Faculty, Department of Mathematics, 25240, Campus, Erzurum, Turkey

^{2}Uludağ University, Education Faculty, Department of Mathematics, 16059, Bursa, Turkey

^{3}Van Yüzüncü Yil University, Faculty of Education, Department of Mathematics Education, Van, Turkey

### Abstract

In this paper, we establish several inequalities for *s*-convex mappings that are connected with the Riemann-Liouville fractional integrals. Our results have some relationships with certain integral inequalities in the literature.

**Keywords:** Hadamard's Inequality, Riemann-Liouville Fractional Integration, Hölder Inequality, s-convexity

Received November 07, 2016; Revised December 19, 2016; Accepted December 27, 2016

**Copyright**© 2017 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Çetin Yildiz, M. Emіn Özdemіr, Havva Kavurmaci Önalan. Fractional Integral Inequalities via
*s*-Convex Functions.*Turkish Journal of Analysis and Number Theory*. Vol. 5, No. 1, 2017, pp 18-22. http://pubs.sciepub.com/tjant/5/1/4

- Yildiz, Çetin, M. Emіn Özdemіr, and Havva Kavurmaci Önalan. "Fractional Integral Inequalities via
*s*-Convex Functions."*Turkish Journal of Analysis and Number Theory*5.1 (2017): 18-22.

- Yildiz, Ç. , Özdemіr, M. E. , & Önalan, H. K. (2017). Fractional Integral Inequalities via
*s*-Convex Functions.*Turkish Journal of Analysis and Number Theory*,*5*(1), 18-22.

- Yildiz, Çetin, M. Emіn Özdemіr, and Havva Kavurmaci Önalan. "Fractional Integral Inequalities via
*s*-Convex Functions."*Turkish Journal of Analysis and Number Theory*5, no. 1 (2017): 18-22.

Import into BibTeX | Import into EndNote | Import into RefMan | Import into RefWorks |

### 1. Introduction

Let be a convex function defined on the interval I of real numbers and The following double inequality

is well known in the literature as Hadamard’s inequality. Both inequalities hold in the reversed direction if *f* is concave.

On November 22, 1881, Hermite (1822-1901) sent a letter to the Journal Mathesis. This letter was published in Mathesis 3 (1883, p: 82) and in this letter an inequality presented which is well-known in the literature as Hermite-Hadamard integral inequality. Since its discovery in 1883, Hermite-Hadamard inequality has been considered the most useful inequality in mathematical analysis. Many uses of these inequalities have been discovered in a variety of settings. Moreover, many inequalities of special means can be obtained for a particular choice of the function *f*.

Let real function *f* be defined on some nonempty interval *I* of real line The function *f* is said to be convex on *I* if inequality

holds for all and

In ^{[3]}, *s*-convex functions defined by Orlicz as following.

**Definition 1.** *A function** ** **where** ** **is said to be s**-**convex** **in the **fi**rst sense if*

for all* ** ** **with** ** **and for some **fi**xed** ** **We** **denote by** ** **the class of all **s**-**convex functions*.

**Definition 2.** *A function** ** **where** ** **is said to be s**-**convex in the second sense if*

*for all** ** ** **with** ** **and for some **fi**xed** ** **We** **denote by** ** **the class of all **s**-**convex functions*.

Orlicz defined these class of functions in ^{[3]} and these definitions were used in the theory of Orlicz spaces in ^{[4]} and ^{[5]}. Obviously, one can see that if we choose *s* = 1, both definitions reduced to ordinary concept of convexity.

For several results related to above definitions we refer readers to ^{[2, 6, 7]} and ^{[8]}.

In ^{[6]}, Dragomir and Fitzpatrick proved a variant of Hermite-Hadamard inequality which holds for the *s*-convex functions.

**Theorem 1. ***Suppose that** ** **is an s-convex** function in the second sence, where** ** **and **let** ** ** **If** ** **then the following** inequalities hold:*

(1.1) |

*The constant** ** **is the best possible in the second inequality in (1.1)*.

In ^{[7]}, Kırmacı et al. obtained Hadamard type inequalities which hold for s-convex functions in the second sence. It is given in the next theorem.

**Theorem 2.** *Let** ** ** **be differentiable function on** ** **such that** ** **where** ** ** **If** ** **is s**-**convex on** ** **for some fixed** ** **and** ** **then:*

(1.2) |

In ^{[1]}, Dragomir and Agarwal proved the following inequality.

**Theorem 3.** *Let** ** **be a differentiable mapping on** ** ** **with** ** and let** ** **If the new mapping** ** **is convex on** ** **then the following inequality holds:*

(1.3) |

In ^{[12]}, Set et al. proved the following Hadamard type inequality for *s*-convex functions in the second sense via Riemann-Liouville fractional integral.

**T****heorem 4.** *Let** ** **be a differentiable **mapping on** ** **with** ** **such that** ** **If ** **is s**-**convex in the second sense on** ** **for some fixed** ** **and** ** **then the following inequality for fractional integrals** **holds*

(1.4) |

Now, we give some necessary definitions and mathematical preliminaries of fractional calculus theory which are used throughout this paper, see (^{[9]}).

**Definition 3.** *Let** ** **The Riemann-Liouville integrals** ** **and** ** **of order** ** **with** ** **are de**fi**ned by*

*and*

*respectively where** ** **Here is** *

In the case of = 1, the fractional integral reduces to the classical integral. For some recent results connected with fractional integral inequalities see ([10-17]^{[10]}).

In order to prove our main theorems, we need the following lemma:

**Lemma 1. ***(see *^{[18]}*) Let** ** **be a differentiable mapping on** ** **with** ** ** **If** ** **then the following equality for fractional integral holds:*

### 2. Main Results

**Theorem 5.** *Let** ** **be a differentiable **mapping on** ** **with** ** s**uch that** ** **If** ** **is s**-**convex in the second sense on** ** **for some fixed** ** **then the following inequality for fractional integrals holds*

*Proof.* From Lemma 1 and using the properties of modulus, we get

Since is *s*-convex on we obtain the inequality

Hence,

and

We obtain

**Theorem 6.** *Let** ** **be a differentiable mapping on** ** **with** ** **such that** ** **If** ** **is s-convex in the second sense on** ** **for some fixed** ** **and** ** **with** ** **then the following inequality for fractional integrals holds*

*where** *

*Proof.* By Lemma 1 and using Hölder inequality with the properties of modulus, we have

We know that for and

therefore

Since is *s*-convex on , we get

which completes the proof.

**Cor****ollary 1.** *If in Theorem 6, we choose **r **= **b**, **then we have*

(2.1) |

**Remark 1.*** If we choose** ** **ve** ** **in Corollary 6, then we have*

*which is the inequality in (1.3)*.

**T****heorem 7.** *Let** ** **be a di**ff**erentiable **mapping on** ** **with** ** **such that** ** **If** ** **is **s**-**convex in the second sense on** ** **for** **some **fi**xed** ** **and** ** **then the following inequality for fractional integrals** **holds*

*Proof.* From Lemma 1 and using the well-known power mean inequality with the properties of modulus, we have

On the other hand, we have

Since is *s*-convex on , we obtain

and

Since

and

Therefore, we have

**Remark 2.** *If we choose r = b in Theorem 7, we obtain the inequality in (1.4) of Theorem 4.*

**Remark 3.** *If we choose r = b and *α* = 1 in Theorem 7, we obtain the inequality in (1.2) of Theorem 2*.

**Theorem 8.** *Let** ** **be a di**ff**erentiable mapping on** ** **with** ** **such that** ** **If** ** **is **s**-**concave in the second sense on** ** **and** ** **with** ** **then the following inequality for fractional integrals holds*

*Proof. *From Lemma 1 and using Hölder inequality, we have

Since is *s*-concave on , we get

so

which completes the proof.

**Corollary 2.*** If we choose r = b in Theorem 8, we obtain*

### References

[1] | S.S. Dragomir and R.P. Agarwal. Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11(5)(1998), 91-95. | ||

In article | View Article | ||

[2] | H. Hudzik and L. Maligranda. Some remarks on s-convex functions, Aequationes Math., 48 (1994), 100-111. | ||

In article | View Article | ||

[3] | W. Orlicz. A note on modular spaces-I, Bull. Acad. Polon. Sci. Math. Astronom. Phys., 9 (1961), 157-162. | ||

In article | |||

[4] | W. Matuszewska and W. Orlicz. A note on the theory of s-normed spaces of φ-integrable functions, Studia Math., 21 (1981), 107-115. | ||

In article | |||

[5] | J. Musielak. Orlicz spaces and modular spaces, Lecture Notes in Mathematics, Vol. 1034, Springer-Verlag, New York / Berlin, 1983. | ||

In article | View Article | ||

[6] | S.S. Dragomir and S. Fitzpatrick. The Hadamard.s inequality for s-convex functions in the second sense, Demonstratio Math., 32 (4) (1999), 687-696. | ||

In article | |||

[7] | U.S. Kırmacı, M.K. Bakula, M.E. Özdemir and J. Pečarić. Hadamard-type inequalities for s-convex functions, Appl. Math. Comput., 193 (2007), 26-35. | ||

In article | View Article | ||

[8] | S.S. Dragomir, C.E.M. Pearce. Selected topics on Hermite-Hadamard inequalities and applications, RGMIA monographs, Victoria University, 2000. [Online: http://www.staff.vu.edu.au/RGMIA/monographs/hermite-hadamard.html]. | ||

In article | |||

[9] | S.G. Samko, A.A Kilbas and O.I. Marichev. Fractional Integrals and Derivatives Theory and Application, Gordan and Breach Science, New York, 1993. | ||

In article | |||

[10] | M.Z. Sarikaya, E. Set, H. Yaldız and N. Başak. Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57 (9-10) (2013), 2403-2407. | ||

In article | View Article | ||

[11] | M.Z. Sarikaya and H. Ogunmez. On new inequalities via Riemann-Liouville fractional integration, Abst. Appl. Anal., 2012. | ||

In article | |||

[12] | E. Set, M.Z. Sarikaya, M.E. Özdemir and H. Yıldırım. The Hadamard’s inequality for some convex functions via fractional integrals and related results, Jour. of Appl. Math., Statis. Infor., 10 (2) (2014), 69-83. | ||

In article | |||

[13] | S. Belarbi and Z. Dahmani. On some new fractional integral inequalities, J. Ineq. Pure and Appl. Math., 10(3) (2009), Art. 86. | ||

In article | |||

[14] | Z. Dahmani. New inequalities in fractional integrals, International Journal of Nonlinear Science, 9(4) (2010), 493-497. | ||

In article | |||

[15] | Z. Dahmani, On Minkowski and Hermite-Hadamard integral inequalities via fractional integration, Ann. Funct. Anal. 1(1) (2010), 51-58. | ||

In article | View Article | ||

[16] | Z. Dahmani, L. Tabharit, S. Taf. Some fractional integral inequalities, Nonl. Sci. Lett. A., 1(2) (2010), 155-160. | ||

In article | |||

[17] | Z. Dahmani, L. Tabharit, S. Taf. New generalizations of Grüss inequality using Riemann-Liouville fractional integrals, Bull. Math. Anal. Appl., 2(3) (2010), 93-99. | ||

In article | |||

[18] | M.E. Özdemir, S.S. Dragomir and Ç. Yıldız. The Hadamard’s inequality for convex function via fractional integrals, Acta Math. Sci., 2013, 33B (5), 1293-1299. | ||

In article | View Article | ||