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 |
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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.
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