1. Introduction
Convexity plays an important role in many features of mathematical programming including, for example, sufficient optimality conditions and duality theorems. The topic of convex functions has been treated extensively in the classical book by Hardy, Littlewood and Polya [
1]. The study of fractional order derivatives and integrals is called fractional calculus. Fractional calculus have important applications in all fields of applied sciences. Fractional integration and fractional differentiation appear as basic part in the subject of partial differential equations [
2,
3]. Many types of fractional integral as well as differential operators have been defined in literature. Classical Caputo-fractional derivatives were introduced by Michele Caputo in [
4] in 1967. Toader [
5] defined the -convexity as follows:
Definition 1.
The function is said to be convex, if we have%
for all and
Definition 2.(see[6])
The function is exponential-convex, if
for alland and .
Definition 3.(see[7])
The function is -convex in second sense with if
for alland and
Definition 4.(see[8])
The function is exponential -convex in second sense with if
for\ alland\ and
Definition 5.(see[9])
The function is -convex in second sense with and be an interval, if
for alland .
Definition 6.
The function is exponential -convex in second sense with and be an interval, if
for alland\ and
The previous era of fractional calculus is as old as the history of differential calculus.
They generalize the differential operators and ordinary integral. However, the fractional derivatives have some basic properties than the corresponding classical ones.
On the other hand, besides the smooth requirement, Caputo derivative does not coincide with the classical derivative [
10]. We give the following definition of Caputo fractional derivatives, see [
2,
11,
12,
13].
Definition 7.
let be a space of functions having derivatives absolutely continuous with and , .
The right sided Caputo fractional derivative is as follows:
The left sided caputo fractional derivative is as follows:
The Caputo fractional derivative coincides with whereas coincides with with exactness to a constant multiplier , if and usual derivative of order exists.
In particular. we have
where and .
In this paper, we establish several new integral inequalities including
Caputo fractional derivatives for exponential -convex functions. By using convexity for exponential -convex functions of any positive integer order differentiable function some novel results are given. The purpose of this paper is to introduce some fractional inequalities for the Caputo-fractional derivatives via -convex functions which have derivatives of any integer order.
2. Main Results
First we give the following estimate of the sum of left and right handed Caputo fractional derivatives.
Theorem 1.
Let be a real valued -time differentiable function where is a positive integer. If is a positive -convex function, then for and , the following inequality for Caputo fractional derivatives holds:
Proof.
Let us consider the function on the interval and is a positive integer. For and , the following inequality holds:
Since is exponential -convex therefore for , we have
Multiplying inequalities (5) and (6), then integrating with respect to over , we have
Now we consider function on the interval . For , the following inequality holds:
Since is exponential -convex on , therefore for , we have
Multiplying inequalities (8) and (9), then integrating with respect to over , we have
Adding (7) and (10) we get the required inequality in (4).
Corollary 1.
By setting in (4) we get the following fractional integral inequality:
Remark 1.
By setting the inequality will be of the form:
Remark 2.
By setting , , and , we will get Corollary 2.1 of [14].
Now, we give the next result stated in the following theorem.
Theorem 2.
Let be a real valued -time differentiable function where is a positive integer. If is exponential (s,m)-convex function, then for , the following inequality for Caputo fractional derivatives holds
Proof.
Since is exponential -convex function and is a positive integer, therefore for and , we have
from which we can write
We consider the second inequality of inequality (14)
Now for , we have
The product of last two inequalities give
Integrating with respect to over , we have
and
Therefore (17) takes the form:
If one consider from (14) the first inequality and proceed as we did for the second inequality, then following inequality can be obtained:
From (18) and (19), we get
On the other hand, for , using convexity of as a exponential -convex function, we have
Also for and , we have
By adopting the same treatment as we have done for (14) and (16) one can obtain from (21) and (22) the following inequality:
By combining the inequalities (20) and (23) via triangular inequality we get the required inequality.
It is interesting to see the following inequalities as a special case.
Corollary 2.
By setting in (13), we get the following fractional integral inequality:
Remark 3.
By setting the inequality will be of the form,
Remark 4.
By setting , , and , we will get Corollary 2.2 of [14].
Before going to the next theorem we observe the following result.
Lemma 1.
Let , be a exponential (s,m)-convex function. If is exponentially symmetric about , then the following inequality holds
Proof.
Since is exponential (s,m)-convex we have
Since is symmetric about , therefore we get .
By substituting where , we get
Also is exponentially symmetric about , therefore we have and inequality in (24) holds.
Theorem 2.
Let be a real valued -time differentiable function where is a positive integer. If is a positive exponential (s,m)- convex and symmetric about , then for and , the following inequality for Caputo fractional derivatives holds
where for and for .
Proof.
For , we have
Also is exponential -convex function, we have
Multiplying (28) and (29) and then integrating with respect to over , we have
From which we have
On the other hand for we have
Multiplying (29) and (31) and then integrating with respect to over , we get
From which we have
Adding (30) and (32) we get the second inequality.
Since is exponential s-convex and symmetric about using Lemma , we have
Multiplying with on both sides and then integrating over , we have
By definition of Caputo fractional derivatives for exponential -convex function, one can have
Multiplying (33) with , then integrating over , one can get
Adding (35) and (36), we get the first inequality.
Corollary 3.
If we put in (27), then we get
\noindent where for and for .
Remark
By setting , and in Theorem 3 we will get Theorem 2.3 of [14].
Autho Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Conflict of Interests
The authors declare no conflict of interest.