Fractional Integral Inequalities via s-Convex Functions

Çetin Yildiz, M. Emіn Özdemіr, Havva Kavurmaci Önalan

Turkish Journal of Analysis and Number Theory

Fractional Integral Inequalities via s-Convex Functions

Çetin Yildiz1,, M. Emіn Özdemіr2, Havva Kavurmaci Önalan3

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

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

3Van 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.

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.

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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 first sense if

for all with and for some fixed 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 fixed 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.

Theorem 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 defined 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 such 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.

Corollary 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).

Theorem 7. 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

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 differentiable 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-Hadamards 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 Hadamards 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
 
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