On refinements of Hermite-Hadamard type inequalities for Riemann-Liouville fractional integral operators

Article History: Received 02 March 2018 Accepted 10 December 2018 Available 30 January 2019 In this paper, we first establish weighted versions of Hermite-Hadamard type inequalities for Riemann-Liouville fractional integral operators utilizing weighted function. Then we obtain some refinements of these inequalities. The results obtained in this study would provide generalization of inequalities proved in earlier works.


Introduction
The Hermite-Hadamard inequality, which is the first fundamental result for convex mappings with a natural geometrical interpretation and many applications, has drawn attention much interest in elementary mathematics.The inequalities discovered by C. Hermite and J. Hadamard for convex functions are considerable significant in the literature (see, e.g., [17, p.137], [2]).These inequalities state that if f : I → R is a convex function on the interval I of real numbers and a, b ∈ I with a < b, then Both inequalities hold in the reversed direction if f is concave.
The remainder of this work is organized as follows: we first give the definitions of Riemann-Liouville fractional integrals and present some Hermite-Hadamard type inequalities for Riemann-Liouville fractional integral operators in Section 2. In the main section, we first establish a new weighted version of Hermite-Hadamard inequality for Riemann-Liouville fractional integrals.Moreover, we obtain some refinements of this result using the symmetric weighted function.We give also some special cases of these inequalities.In the last section, we give some conclusions and future directions of research.

Preliminaries
In the following we will give some necessary definitions and mathematical preliminaries of fractional calculus theory which are used further in this paper.
. It is remarkable that Sarikaya et al. [20] first give the following interesting integral inequalities of Hermite-Hadamard type involving Riemann-Liouville fractional integrals.
, then the following inequalities for fractional integrals hold: with α > 0.

Now we give the following lemma:
Lemma 1. [22,25] Let f : [a, b] → R be a convex function and h be defined by In [22], Xiang obtained following important inequalities for the Riemann-Liouville fractional integrals utilizing the Lemma 1: then W H is convex and monotonically increasing on [0, 1] and with α > 0 where , then W P is convex and monotonically increasing on [0, 1] and with α > 0 where In this study, we establish some refinements of Hermite-Hadamard type inequalities utilizing fractional integrals which generalize the inequalities ( 2), ( 3) and (4).

Refinements of Hermite Hadamard Type Inequalities
In this section, we will present refinements of Hermite-Hadamard type inequalities via Riemann-Liouville fractional integral operators .
The following Lemma will be frequently used to prove our results. .Then, we have the following inequality and if the function w is monotonic on [a, b] , then we have with α > 0.
Proof.By the hypothesis of symmetricity of the function w, we have and we also have ≤ f (w(s)) + f (w(a + b − s)) .
Multiplying by (s−a) α−1 both sides of (7) and integrating with respect to s on [a, b], we deduce that which completes the proof of the inequality (5).By the monotonicity w, we have Multiplying both sides of (8) by (s−a) α−1
Remark 1.If we choose w(t) = t in Theorem 4, then the inequalities ( 5) and ( 6) reduce to left and right hand sides of the inequality (2), respectively., i.e. 1  2 [w(s) + w(a then W H w is convex and monotonically increasing on [0, 1] and we have the following inequalities with α > 0 where Since f is convex, we have Hence, we get W H w is convex on [0, 1] .On the other hand, we have Let . By the symmetricity of the function w, we have Hence, applying Lemma 2, we have Multiplying both sides of (11) by and integrating with respect to s on a, a+b 2 , then by considering the equality (10), we deduce that W H w (t 1 ) ≤ W H w (t 2 ).Thus, W H w is monotonically increasing on [0, 1] .Using the facts that then we obtain the desired result.Thus, the proof is completed.
Remark 3. If we choose w(t) = t in Theorem 5, then the inequality (9) reduces to the inequality (3).
= f (w (a)) + f (w (b)) 2 with α > 0 where Proof.By the way similar to in Theorem, it can be easily proved by convexity of f that W P w is convex on [0, 1] .Using change of variable, we have and w is monotonic, we have for s ∈ [a, b] .By the equality (14) and the inequality (15), we have and integrating with respect to s on a, a+b 2 , then by considering the equality (13), we deduce that W P w (t 1 ) ≤ W P w (t 2 ).Hence, W P w is monotonically increasing on [0, 1] .This completes the proof.

Conclusion
In this paper, we present some new weighted refinements of Hermite-Hadamard inequalities for Riemann-Liouville fractional integrals.For further studies we propose to consider the Hermite-Hadamard type inequalities for other fractional integral operators Let f : [a, b] → R be convex function.Then the inequality holds, where g : [a, b] → R is nonnegative, integrable and symmetric to (a + b)/2.

Theorem 4 ., w a+b 2 ,
Let f : [a, b] → R be convex function with a < b and f ∈ L [a, b] .Let the weight function w : [a, b] → R be continuous and symmetric about the point a+b 2 i.e. 1 2 [w(s) + w(a + b − s)] = w a+b 2 |w(s) − w(a + b − s)| ≤ |w(a) − w(b)| for s ∈ [a, b] and by symmetricity of the function w, we have w(s) + w(a + b − s) = w(a) + w(b) for s ∈ [a, b] .Applying Lemma 2, we get

Remark 2 .Theorem 5 .
If we choose α = 1 in Theorem 4, then Theorem 4 reduces to Theorem 1 proved in [9].Let the weight function w : [a, b] → R be continuous and symmetric about the point a+b 2 , w a+b 2

Remark 4 .Theorem 6 ., w a+b 2 ,
If we choose α = 1 in Theorem 5, then Theorem 5 reduces to Theorem 2 proved in[9].Let the weight function w : [a, b] → R be continuous and monotonic on [a, b] and let w be symmetric about the point a+b 2 i.e.

1 2 2 .
[w(s) + w(a + b − s)] = w a+b If f : [a, b] → R is a convex function on [a, b],then W P w is convex and monotonically increasing on [0, 1] and we have the following inequalities