Generalized Ostrowski type inequalities for functions whose local fractional derivatives are generalized s-convex in the second sense

• BibTex
• RIS
• APA
• Harvard
• IEEE
• MLA
• Vancouver
• Chicago

@article{Budak_2016,
   doi = {10.17512/jamcm.2016.4.02},
   url = {https://doi.org/10.17512/jamcm.2016.4.02},
   year = 2016,
   publisher = {The Publishing Office of Czestochowa University of Technology},
   volume = {15},
   number = {4},
   pages = {11--21},
   author = {Hüseyin Budak and  Mehmet Zeki Sarikaya and  Erhan Set},
   title = {Generalized Ostrowski type inequalities for functions whose local fractional derivatives are generalized s-convex in the second sense},
   journal = {Journal of Applied Mathematics and Computational Mechanics}
}

TY  - JOUR
DO  - 10.17512/jamcm.2016.4.02
UR  - https://doi.org/10.17512/jamcm.2016.4.02
TI  - Generalized Ostrowski type inequalities for functions whose local fractional derivatives are generalized s-convex in the second sense
T2  - Journal of Applied Mathematics and Computational Mechanics
JA  - J Appl Math Comput Mech
AU  - Budak, Hüseyin
AU  - Sarikaya, Mehmet Zeki
AU  - Set, Erhan
PY  - 2016
PB  - The Publishing Office of Czestochowa University of Technology
SP  - 11
EP  - 21
IS  - 4
VL  - 15
SN  - 2299-9965
SN  - 2353-0588
ER  - 

Budak, H., Sarikaya, M., & Set, E. (2016). Generalized Ostrowski type inequalities for functions whose local fractional derivatives are generalized s-convex in the second sense. Journal of Applied Mathematics and Computational Mechanics, 15(4), 11-21. doi:10.17512/jamcm.2016.4.02

Budak, H., Sarikaya, M. & Set, E., 2016. Generalized Ostrowski type inequalities for functions whose local fractional derivatives are generalized s-convex in the second sense. Journal of Applied Mathematics and Computational Mechanics, 15(4), pp.11-21. Available at: https://doi.org/10.17512/jamcm.2016.4.02

[1]H. Budak, M. Sarikaya and E. Set, "Generalized Ostrowski type inequalities for functions whose local fractional derivatives are generalized s-convex in the second sense," Journal of Applied Mathematics and Computational Mechanics, vol. 15, no. 4, pp. 11-21, 2016.

Budak, Hüseyin, Mehmet Zeki Sarikaya, and Erhan Set. "Generalized Ostrowski type inequalities for functions whose local fractional derivatives are generalized s-convex in the second sense." Journal of Applied Mathematics and Computational Mechanics 15.4 (2016): 11-21. CrossRef. Web.

1. Budak H, Sarikaya M, Set E. Generalized Ostrowski type inequalities for functions whose local fractional derivatives are generalized s-convex in the second sense. Journal of Applied Mathematics and Computational Mechanics. The Publishing Office of Czestochowa University of Technology; 2016;15(4):11-21. Available from: https://doi.org/10.17512/jamcm.2016.4.02

Budak, Hüseyin, Mehmet Zeki Sarikaya, and Erhan Set. "Generalized Ostrowski type inequalities for functions whose local fractional derivatives are generalized s-convex in the second sense." Journal of Applied Mathematics and Computational Mechanics 15, no. 4 (2016): 11-21. doi:10.17512/jamcm.2016.4.02

GENERALIZED OSTROWSKI TYPE INEQUALITIES FOR FUNCTIONS WHOSE LOCAL FRACTIONAL DERIVATIVES ARE GENERALIZED S-CONVEX IN THE SECOND SENSE

Hüseyin Budak ¹, Mehmet Zeki Sarikaya ², Erhan Set ³

^1,2 Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce-Turkey
³ Department of Mathematics, Faculty of Arts and Sciences, Ordu University, 52200, Ordu, Turkey
hsyn.budak@gmail.com, sarikayamz@gmail.com, erhanset@yahoo.com
Received: 14 March 2016; accepted: 15 September 2016

Abstract. In this paper, we establish some generalized Ostrowski type inequalities for functions whose local fractional derivatives are generalized s-convex in the second sense.

Keywords: generalized Hermite-Hadamard inequality, generalized Hölder inequality, generalized convex functions

1. Introduction

In 1938, Ostrowski established the following interesting integral inequality for differentiable mappings with bounded derivatives [1]:

Theorem 1. (Ostrowski inequality) Let be a differentiable mapping on whose derivative is bounded on i.e. Then, we have the inequality

(1)

for all . The constant is the best possible.

In recent years, the fractal theory has received significant attention. The calculus on the fractal set can lead to better comprehension for the various real world models from science and engineering [2-19].

The purpose of this paper is to establish some local fractional integral inequalities using generalized s-convex in the second sense on real linear fractal set . This paper is divided into the following three sections. In Section 2, we give the definitions of the local fractional derivatives and local fractional integrals and introduce several useful notations on fractal space which will be used our main results. In Section 3, the main results are presented.

2. Preliminaries

Recall the set of real line numbers and use the Gao-Yang-Kang's idea to describe the definition of the local fractional derivative and local fractional integral, see [14, 15] and so on.

Recently, the theory of Yang’s fractional sets [yang] was introduced as follows.

For we have the following -type set of element sets:

The -type set of integer is defined as the set

The -type set of the rational numbers is defined as the set

The -type set of the irrational numbers is defined as the set

The -type set of the real line numbers is defined as the set

If and belongs the set of real line numbers, then

(1) and belongs the set

(2)

(3)

(4)

(5)

(6)

(7) and

The definition of the local fractional derivative and local fractional integral can be given as follows.

Definition 1. [14] A non-differentiable function is called to be local fractional continuous at , if for any there exists such that

holds for where If is local continuous on the interval we denote

Definition 2. [14] The local fractional derivative of of order at is defined by

where .

If there exists for any then we denoted where .

Definition 3. [14] Let Then the local fractional integral is defined by,

with and where and is a partition of interval

Here, it follows that if and if If for any there exists then we denote by

Lemma 1. [14]

(1) (Local fractional integration is anti-differentiation) Suppose that then we have

(2) (Local fractional integration by parts) Suppose that and then we have

Lemma 2. [14]

Lemma 3. (Generalized Hölder’s inequality) [14] Let with then

In [7], the authors introduced two kinds of generalized s-convex functions on fractal sets as follows:

Definition 4. Let A function is said to be generalized s-convex in the first sense, if

for all and with We denote this by

Definition 5. A function is said to be generalized s-convex in the second sense, if

for all and with We denote this by

If we have the reverse inequality, then is called s-concave.

Sarikaya and Budak proved the following generalized Ostrowski inequality in [10]:

Theorem 2. (Generalized Ostrowski inequality) Let be an interval, ( is the interior of ) such that and for with Then, for all we have the identity

(2)

In [8], Mo and Sui established the following Hermite-Hadamard inequality for generalized s-convex functions on a real linear fractal set

Theorem 3. Suppose that is a generalized s-convex function in the second sense, where . Let , . If , then the following inequalities hold:

If is a generalized s-concave, then we have the reverse inequality.

3. Main results

We will start with a generalized identity for local fractional integrals:

Theorem 4. Let be an interval, ( is the interior of ) such that and for with . Then, we have the identity

(3)

for all

Proof. Using the local fractional integration by parts (Lemma 1), we have

(4)

Similarly, we have

(5)

Using (4) and (5), we obtain

which is the required result.

Theorem 5. The assumptions of Theorem 4 are satisfied. If is generalized s-convex in the second sense on for some fixed , then we have the inequality

(6)

for all where

Proof. By Theorem 4 and since is generalized s-convex in the second sense, then we have

Here, we used the fact

and

This completes the proof.

Remark 1. If we take in (6), then (6) reduces to (2).

Corollary 1. Under assumption of Theorem 5 with we have the following midpoint inequality

Theorem 6. The assumptions of Theorem 4 are satisfied. If is generalized s-convex in the second sense on for some fixed , then we have the inequality

(7)

for all where with

Proof. Taking modulus in (3) and using the generalized Hölder's inequality (Lemma 3), we have

Since is generalized s-convex in the second sense on , then we have

(8)

and similarly,

(9)

If we substitute the inequality (8) and (9), then we obtain the desired result.

Corollary 2. Under assumption of Theorem 6 with we have the following midpoint inequality

Theorem 7. The assumptions of Theorem 4 are satisfied. If is generalized s-concave on for some fixed , then we have the inequality

(10)

for all where with

Proof. From Theorem 4 and using generalized Hölder's inequality, we have

Since is generalized s-concave on applying Theorem 3, we have

(11)

and

(12)

If we substitute the inequality (11) and (12), then we obtain the desired result.

Corollary 3. Under assumption of Theorem 7 with we have the following midpoint inequality

where with

4. Conclusions

In this paper, we presented some Ostrowski type inequalities for function whose local fractional derivatives are generalized s-convex in the second sense. A further study could be assess similar inequalities by using different types of kernels or convexity.

References

[1] Budak H., Sarikaya M.Z., Yildirim H., New inequalities for local fractional integrals, Iranian Journal of Science and Technology (Sciences), in press.

[2] Chen G-S., Generalizations of Hölder's and some related integral inequalities on fractal space, Journal of Function Spaces and Applications Volume 2013, Article ID 198405.

[3] Kılıçman A., Saleh W., Notions of generalized s-convex functions on fractal sets, Journal of Inequalities and Applications 2015, 312. DOI 10.1186/s13660-015-0826-x.

[4] Mo H., Sui X., Yu D., Generalized convex functions on fractal sets and two related inequalities, Abstract and Applied Analysis 2014, Article ID 636751, 7 pages.

[5] Mo H., Generalized Hermite-Hadamard inequalities involving local fractional integrals, arXiv:1410.1062 [math.AP].

[6] Mo H., Sui X., Generalized s-convex function on fractal sets, arXiv:1405.0652v2 [math.AP].

[7] Mo H., Sui X., Hermite-Hadamard type inequalities for generalized s-convex functions on real linear fractal set R^α(0 < α < 1), arXiv:1506.07391v1 [math.CA].

[8] Ostrowski A.M., Über die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, Comment. Math. Helv. 1938, 10, 226-227.

[9] Saleh W., Kılıçman A., On generalized s-convex functions on fractal set, JP Journal of Geometry and Topology 2015, 17(1), 63-82.

[10] Sarikaya M.Z., Budak H., Generalized Ostrowski type inequalities for local fractional integrals, RGMIA Research Report Collection 2015, 18, Article 62, 11 p.

[11] Sarikaya M.Z., Erden S., Budak H., Some generalized Ostrowski type inequalities involving local fractional integrals and applications, Advances in Inequalities and Applications 2016, 6.

[12] Sarikaya M.Z., Erden S., Budak H., Some integral inequalities for local fractional integrals, RGMIA Research Report Collection 2015, 18, Article 65, 12 p.

[13] Sarikaya M.Z., Budak H., Erden S., On new inequalities of Simpson's type for generalized convex functions, RGMIA Research Report Collection 2015, 18, Article 66, 13 p.

[14] Yang X.J., Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York 2012.

[15] Yang J., Baleanu D., Yang X.J., Analysis of fractal wave equations by local fractional Fourier series method, Adv. Math. Phys. 2013, Article ID 632309.

[16] Yang X.J., Local fractional integral equations and their applications, Advances in Computer Science and its Applications (ACSA) 2012, 1(4).

[17] Yang X.J., Generalized local fractional Taylor's formula with local fractional derivative, Journal of Expert Systems 2012, 1(1), 26-30.

[18] Yang X.J., Local fractional Fourier analysis, Advances in Mechanical Engineering and its Applications 2012, 1(1), 12-16.

[19] Yang X.J., Baleanu D., Srivastava H.M., Local Fractional Integral Transforms and their Applications, Elsevier, 2016.