On Hermite-Hadamard-Fejer type Inequality for Convex Functions via Fractional Integrals

In this paper, we have established some generalized integral inequalities of Hermite-Hadamard-Fej\'er type for generalized fractional integrals. The results presented here would provide generalizations of those given in earlier works.


Introduction
Let f : I ⊂ R → R be a convex function define on an interval I of real numbers, and a, b ∈ I with a < b. Then the following inequalities hold: It was first discovered by Hermite in 1881 in the Journal Mathesis. This inequality (1.1) was nowhere mentioned in the mathematical literature untill 1893. In [4], Beckenbach, a leading expert on the theory of convex functions, wrote that the inequality (1.1) was proved by Hadamard in 1893. In 1974, Mitrinovič found Hermite and Hadamard's note in Mathesis. That is why, the inequality (1.1) was known as Hermite-Hadamard inequality. We note that Hermite-Hadamard's inequality may be regarded as a refinements of the concept of convexity and it follows easily from Jensen's inequality. This inequality (1.1) has been received renewed attention in recent years and a remarkable variety of refinements and generalizations have been found in [4]- [16], [18], [20].
The most well known inequalities connected with the integral mean of a convex functions are Hermite-Hadamard inequalities or its weighted versions, the so-called Hermite-Hadamard-Fejěr inequality.
In [10], Fejér established the following Fejér inequality which is the weighted generalization of Hermite-Hadamard inequalities (1.1): Theorem 1. Let f : I → R be a convex on I and let a, b ∈ I with a < b. Then the inequality holds, where f : [a, b] → R is nonnegative, integrable, and symmetric to a + b 2 .
] . If f is a convex function on [a, b] , then the following inequalities for fractional integrals holds with α > 0.
In [5] Set et. al. obtained the following lemma.
, then the following identity for fractional integrals holds: We give some necessary definitions and mathematical preliminaries of fractional calculus theory which are used throughout this paper.
In this paper, we have established some generalized fractional integral inequalities. The results presented here would provide generalizations of those given in earlier works.

Main Results
, then the following identity for fractional integrals holds: Proof. It suffices to note that By integration by parts, we get and similarly . Thus, can write Multiplying the both sides (Γ (α)) −1 , we obtain (2.1) which complates the proof. Remark 2. If we choose h(x) = x, g(x) = 1 and α = 1 in Lemma 2, we obtain Lemma 2.1 in [21].
, then the following inequality for fractional integrals holds: From Lemma 2, we have This completes the proof.
From Lemma 2, power mean inequality and the convexity of |f ′ | q , it follows that Remark 5. If we choose h(x) = x in Theorem 4, we obtain Theorem 7 in [5].
Lemma 3. Let f : [a, b] → R be a differentiable mapping on (a, b) with a < b and let g : , then the following identity for fractional integrals holds: where Proof. It suffices to note that By integration by parts, we get and similarly Thus, can write Multiplying the both sides (Γ (α)) −1 , we obtain (2.4) which complates the proof.
Remark 7. If we choose h(x) = x and g(x) = 1 in Lemma 3, we have Theorem 5. Let f : I → R be a differentiable mapping on I • and f ′ ∈ X p h [a, b] with a < b and g : [a, b] → R is continuous. If |f ′ | is convex on [a, b] , then the following inequality for fractional integrals holds: