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

### Hüseyin Budak

,### Mehmet Zeki Sarikaya

,### Erhan Set

Journal of Applied Mathematics and Computational Mechanics |
Download Full Text |

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.