Some integral inequalities for m-convex functions via generalized fractional integral operator containing generalized Mittag-Leffler function

In this paper, we are interested to prove some Hadamard and Fejér– Hadamard-type integral inequalities for m-convex functions via generalized fractional integral operator containing the generalized Mittag-Leffler function. In connection with we obtain some known results. Subjects: Science; Mathematics & Statistics; Advanced Mathematics; Pure Mathematics


Introduction
Convex functions play an important role in the study of mathematical analysis. A close generalization of convex functions is m-convex function introduced by Toader (1984). where a, b ∈ I with a < b.

PUBLIC INTEREST STATEMENT
Inequalities are very useful almost in all areas of Mathematics. Fractional integral inequalities are useful in establishing the uniqueness of solutions of certain partial differential equations also provides upper and lower bounds for the solutions of fractional boundary value problems. In this paper, we have established fractional integral inequalities of Hadamard and Fejer-Hadamard type using m-convex functions. Also we have deduced some known results.
and Salim and Faraj (2012), Srivastava and Tomovski (2009) properties of generalized integral operator and generalized Mittag-Leffler function have been studied in brief. Salim and Faraj (2012) it is proved that E , ,k , ,l (t) is absolutely convergent for all t where k < l + .
Since we say that We use this definition of S in sequel in our results.
In Farid (2016) the Hadamard and the Fejér-Hadamard inequality for generalized fractional integral operator containing Mittag-Leffler function defined in (1) are proved. The Hadamard and the Fejér-Hadamard-type inequality for several fractional integral operators are also mentioned in this paper. Also in Farid, Rehman, and Zahra (2016), Iscan (2015), Noor, Noor, and Awan (2015) authors proved the Hadamard and the Fejér-Hadamard-type inequalities for Riemann-Liouville fractional integral operator (Chen & Katugampola, 2017).
In Mubeen and Habibullah (2012) the Riemann-Liouville k-fractional integral operator is defined, we have obtained some results for this operator.
Definition 3 Let f ∈ L 1 [a, b]. Then k-fractional integrals of order , k > 0 with a ≥ 0 are defined as: and where Γ k ( ) is the k-Gamma function defined as: One can note that In this paper, we give the Hadamard and the Fejér-Hadamard-type inequalities for m-convex function via generalized fractional integral operator containing generalized Mittag-Leffler function. (3) The results of Dragmoir and Agarwal (1998), Farid (2016), Iscan (2015), Noor et al. (2015), Sarikaya, Set, Yaldiz, and Basak (2013) are special cases of our results. It is also remarked that many integral inequalities for different kinds of integral operators can be obtained.

Main results
First we prove the following lemmas.
Lemma 2.1 Let g: [a, b] → ℝ be an integrable and symmetric about a+mb 2 and g ∈ L [a, b]. Then we have . By the definition of generalized fractional integral operator containing Mittag-Leffler function, we have replace x by a + mb − x in Equation (6) 2 .
(8) Using Lemma 2.1, we have In the same way we have Adding (9) and (10) we get (8). ✷ Using Lemma 2.2 we prove the following theorem. [a, b] and g: [a, b] → ℝ is continuous and symmetric about a+mb 2 , then for k < l + the following inequality holds where ‖g‖ ∞ = sup t∈ [a,b] �g(t)�.
Proof Using Lemma 2.2 we have Using symmetry of g one can have After simple calculation one can have and Using the above calculations in (14) we have This completes the proof. ✷ In the following corollary, we have the Fejér-Hadamard-type inequality for the k-fractional Riemann-Liouville integral operator (Sarikaya & Kuraca, 2014).
Corollary 2.4 In Theorem 2.3 if we put = 0, = k and m = 1 then we have the following inequality , k > 0.
In the following corollary we obtain the Fejér-Hadamard-type inequality for Riemann-Liouville fractional integral operator.
Proof Using Lemma 2.2, Hölder inequality, (13) and m-convexity of |f ′ | q respectively we have [a,b] �g(t)�, and absolute convergence of Mittag-Leffler function, inequality (16) becomes After simplification, we get Using (17) in above inequality, we have After integrating and simplifying above inequality we get (15). ✷ In the following we get an inequality for the k-fractional Riemann-Liouville integral operator.