k-fractional integral inequalities of Hadamard type for exponentially (s;m)-convex functions

The aim of this article is to present fractional versions of the Hadamard type inequalities for exponentially (s,m)-convex functions via k-analogue of Riemann-Liouville fractional integrals. The results provide generalizations of various known fractional integral inequalities. Some special cases are analyzed in the form of corollaries and remarks.


Introduction
Fractional integral inequalities are useful generalizations of classical inequalities. The Hadamard inequality is the geometric interpretation of convex functions which has been analyzed by many researchers for fractional integral and differentiation operators. For fractional versions of the Hadamard inequality we refer the researchers to [1][2][3][4][5][6][7][8][9]. Convex functions proved very useful for the establishment of new inequalities which have interesting consequences in the theory of classical inequalities. The Hadamard inequality is the most classical inequality for convex functions which is stated in the undermentioned theorem: Theorem 1. [9] If f : I → R is a convex function on the interval I of real numbers and a, b ∈ I with a < b, then In recent years the theory of mathematical inequalities is analyzed via fractional integral operators of different kinds (see, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] and references in there). Inequalities have a significant role in the field of convex analysis, while the classical Hadamard inequality is equivalent to the definition of convex functions. Definition 1. A function f : I → R, where I is an interval in R, is said to be convex function if holds for all x, y ∈ I and r ∈ [0, 1].
The well known beta function is frequently used in the presented results, defined as follows: The beta function of two variables x and y are define as: The objective of this article is to obtain k-fractional integral inequalities for a generalized class of convex functions namely exponentially (s, m)-convex functions. The classical fractional integral operators namely Riemann-Liouville (RL) fractional integrals are defined as follows: respectively. Here Γ(α) is the gamma function and I 0 In [24], Mubeen and Habibullah gave the Riemann-Liouville k-fractional integrals as follows: Then RL k-fractional integrals I α,k a + f and I α,k b − f of order α ∈ C, Re(α) > 0 of f are defined by . In Section 2, we prove k-fractional integral inequality of Hadamard type for exponentially (s, m)convex functions and deduce some related results. In Section 3, we prove a version of k-fractional integral inequality of Hadamard type for differentiable functions f so that | f | is exponentially (s, m)convex. In Section 4, we give some particular cases of results given in Sections 2 & 3.

Main results
In the undermentioned theorem, we give k-fractional integral inequality of Hadamard type for exponentially (s, m)-convex functions.
Proof. Since f is an exponentially (s, m)-convex function, we have Now, if we let w = (1 − r) a m + rb and z = m(1 − r)b + ra in right hand side of above inequality, we get Further, it gives the following inequality which provide the first inequality of (2.1): On the other hand by using exponentially (s, m)-convexity of f , we have By multiplying both sides of above inequality with α 1 2 s r α k −1 and integrating over [0, 1], after some calculations we get By using definition of the beta function, from aforementioned inequality the second inequality of (2.1) is obtained.
In the following we give consequences of above theorem: Corollary 1. The undermentioned inequality holds for exponentially (s, m)-convex functions via RL fractional integrals Proof. By setting k = 1 in inequality (2.1) of Theorem 2, we get the above inequality (2.2).
Corollary 2. The undermentioned result holds for convex functions via RL k-fractional integrals Proof. By setting η = 0, s = 1 and m = 1 in (2.1) of Theorem 2, we get the above inequality (2.3) which is given in [3].

Bounds of Hadamard inequality
In this section k-fractional integral inequalities of Hadamard type for exponentially (s, m)-convex function in terms of the first derivatives has been obtained. For the proof of next result we will use the undermentioned lemma. . If | f | is an exponentially (s, m)-convex function with m ∈ (0, 1], η ∈ R, q > 1. Then for RL k-fractional integrals we have By using exponentially (s, m)-convexity of | f | we will get Now, by using Holder inequality, one has By using the above inequalities in the right hand side of (3.2), we have Corollary 4. The undermentioned inequality holds for exponentially (s, m)-convex functions of Riemann-Liouville fractional integrals Proof. By setting k = 1 in inequality (3.1) of Theorem 3 we get the above inequality (3.3).

Results for (s, m)-convex functions
In this section we discuss some particular cases of the results established in Sections 2 and 3.
. Then we will have the undermentioned inequality: Proof. Its proof is alike to the proof of Theorem 2 or directly (4.1) can be obtained from (2.1) by taking η = 0.

Corollary 5. The undermentioned inequality holds for m-convex functions via RL k-fractional integrals
Proof. By setting s = 1 in inequality (4.1) of Theorem 4 we get the above inequality (4.2).
Corollary 6. The undermentioned inequality holds for s-convex functions via RL k-fractional integrals where 1 p + 1 q = 1. Proof. Its proof is alike to the proof of Theorem 3, or directly (4.4) can be obtained from (3.1) by taking η = 0.
Corollary 7. The undermentioned inequality holds for m-convex functions via RL k-fractional integrals Proof. By setting s = 1 in inequality (4.4) of Theorem 5, we get the above inequality (4.5).
Corollary 8. The undermentioned inequality holds for s-convex functions via RL k-fractional integrals (4.6) Proof. By setting m = 1 in inequality (4.4) of Theorem 5, we get the above inequality (4.6).
Corollary 9. The undermentioned inequality holds for convex functions via RL k-fractional integrals Proof. By setting m = 1 and s = 1 in inequality (4.4) of Theorem 5 we get the above inequality (4.7).

Conclusions
In this article we have presented fractional versions of the Hadamard inequality for exponentially (s, m)-convex functions. By applying definitions of exponentially (s, m)-convex function and Riemann-Liouville fractional integrals we have obtained Hadamard type inequalities in different forms. An identity is used to get error estimations of these Hadamard inequalities. Connections of the results of this paper with already known results are also established. In our future work we are finding the refinements of fractional integral inequalities.