Modification of certain fractional integral inequalities for convex functions

We consider the modified Hermite–Hadamard inequality and related results on integral inequalities, in the context of fractional calculus using the Riemann–Liouville fractional integrals. Our results generalize and modify some existing results. Finally, some applications to special means of real numbers are given. Moreover, some error estimates for the midpoint formula are pointed out.


Introduction
The generalization of certain integral inequalities to the fractional scope, in both continuous and discrete versions, have attracted many researchers in the recent few years and before [1,19,20]. In this article, our work is devoted to Hadamard-Hermite type for convex functions in the framework of Riemann-Liouville fractional type integrals.
A function g : I ⊆ R → R is said to be convex on the interval I, if the inequality g x + (1 -)y ≤ g(x) + (1 -)g(y) ( 1 ) holds for all x, y ∈ I and ∈ [0, 1]. We say that g is concave if -g is convex.
For convex functions (1), many equalities and inequalities have been established by many authors; such as the Hardy type inequality [3], Ostrowski type inequality [7], Olsen type inequality [8], Gagliardo-Nirenberg type inequality [22], midpoint type inequality [10] and trapezoidal type inequality [14]. But the most important inequality is the Hermite-Hadamard type inequality [6], which is defined by where g : I ⊆ R → R is assumed to be a convex function on I where u, v ∈ I with u < v.
A number of mathematicians in the field of applied and pure mathematics have devoted their efforts to generalizing, refining, finding counterparts of, and extending the Hermite-Hadamard inequality (2) for different classes of convex functions and mappings. For more recent results obtained in view of inequality (2), we refer the reader to [2,4,6,13,16,18].
Recently, in [12], Mehrez and Agarwal obtained a new modification of the Hermite-Hadamard inequality (2); this is given by Furthermore, Mehrez and Agarwal obtained many inequalities in view of inequalities (4); for which we refer the reader to their interesting paper [12]. The aim of this paper is to establish new inequalities of Hermite-Hadamard type for convex functions via Riemann-Liouville fractional integrals.

Preliminary lemmas
In order to obtain our main results, we need some qualities which are stated in the following lemmas. where Lemma 3 Let g : I ⊆ R → R be a differentiable function on I o and g ∈ L 1 [u, v]. If g is a convex function on [u, v]. then for ϑ > 0 we have and Proof From the definition of Riemann-Liouville fractional integral, we have By using the change of the variable x = 3 4 + u+v 4 for ∈ [ 3u-v 3 , 3v-u 3 ], we obtain Since g is convex on [u, v], we have It follows from this and (9) that Again, by using the change of the variable z = 3 2 for ∈ [ 3u-v 2 , 3v-u 2 ], we obtain Replace u by 3u-v 2 and v by 3v-u 2 in the right-hand side of inequality (3) and multiply both sides by 1 2 , we get From (10) and (11), we obtain the desired inequality (7) and from (7) we can easily obtain the inequality (8). These complete the proof of Lemma 3.

Hermite-Hadamard type inequalities
Our main results start from the following theorem.
For q > 1 we use the Hölder inequality and the convexity of |g | q on [ 3u-v 2 , 3v-u 2 ] to obtain Analogously, Using (17) and (18) in (14), we get where Applying the formula for (19) and then using the fact that |x r , we obtain the inequality (12). This completes the proof of Theorem 1.

Corollary 1 With similar assumptions to Theorem
which is obtained by Mehrez and Agarwal in [12,Theorem 1].
Remark 2 In [23], the following inequality has been established: We show an analytical and numerical comparison between the left-hand side of inequalities (20) and (21).
In those cases, comparison does not occur analytically between inequalities (20) and (21).
This tells us the right-hand side of inequality (20) is better than the right-hand side of inequality (21). (c) If the function |g | is decreasing on [ 3u-v 2 , u], and increasing on [v, 3v-u 2 ], then we conclude that the right-hand side of inequality (21) is better than the right-hand side of inequality (20).
2. Suppose that m and n represent the right-hand side of inequalities (20) and (21), and g(x) = e x , then we obtain m = 0.199744 and n = 0.167444 when q = 1 and m = 0.155592 when q = 2. Then we conclude that the right-hand side of inequality (20) is worse than the right-hand side of inequalities (21) when q = 1, but better when q = 2.
Proof Applying Hölder's inequality and the convexity of |g | q , q > 1 on [ 3u-v 2 , 3v-u 2 ], we obtain Since (H 1 -H 2 ) q ≤ H q 1 -H q 2 for each H 1 , H 2 > 0 and q > 1, (23) becomes In a similar manner, we get Using (24) and (25) in (14), we get which proves the first inequality of (22). Applying the fact to the last inequality, we get the second inequality of (22). This completes the proof.
Collecting both of Theorems 1 and 2 we obtain the following corollary.
Now by using the convexity of |g |, we find Substituting (30) into (28) we deduce that the inequality (26) holds true for q = 1. Hence the proof of Theorem 3 is completed.

Corollary 4 With similar assumptions to Theorem
(31) Remark 3 In [12, Theorem 3], Mehrez and Agarwal obtained the following inequality: The right-hand side of (31) confirms the modification of our work compared with (32).

Remark 4 If g (x) is bounded on the interval [ 3u-v 2 , 3v-u 2 ], then Theorem 3 reduces to
for some M ∈ R.

Corollary 5 With similar assumptions to Theorem
(35) Proof The proof of this corollary follows from the facts that .
Remark 5 The right-hand side of inequality (35) confirms the modification of our work compared with the right-hand side of inequality (3.24) in [12,Theorem 4].

Corollary 6
With similar assumptions to Theorem 5 if ϑ = 1, we have Remark 7 The right-hand side of inequality (38) confirms the modification of our work compared with the right-hand side of inequality (3.27) in [12,Theorem 5].

Theorem 6
With similar assumptions to Theorem 3, we have Proof Let q > 1, then, by using the Hölder inequality and the convexity of |g | q on [ 3u-v 2 , 3v-u 2 ], we have Using (40) in (28) we obtain the inequality (39) for q > 1. Now, using the convexity of |g | and the properties of the modulus, we find Substituting (41) into (28) we deduce that the inequality (39) holds true for q = 1. Thus (40), (41) and (28) complete the proof of Theorem 6.

Corollary 7 With similar assumptions to Theorem
Remark 8 The right-hand side of inequality (42) confirms the modification of our work compared with the right-hand side of inequality (3.29) in [12,Theorem 6].
Collecting Theorems 3-6 we obtain the following corollary.
A few results for concave functions will be extended in the following theorems.
where q = p p-1 such that p ∈ R, p > 1.
Proof Applying Hölder's inequality to (28), we get By using the concavity of |g | q on [ 3u-v 2 , 3v-u 2 ] and the integral Jensen's inequality, we get and analogously Thus substituting the obtained results of (45) and (46) in (44), we get (43) as desired.

Applications
In this section some applications are presented to demonstrate the usefulness of our obtained results in the previous sections.

Applications to special means
Let u and v are two arbitrary positive real numbers such that u = v, we consider the following special means [17].
(i) The arithmetic mean: (ii) The inverse arithmetic mean: (iii) The geometric mean: (iv) The logarithmic mean: , u = v.
This completes the proof of (55).

Conclusion
In this paper, we generalized the modified Hermite-Hadamard inequality obtained by Mehrez and Agarwal in [12], it can be found in Lemma 3 and Theorems 1-6. Corollaries 4-7 confirm that our results modified the existing results of [12]. Furthermore, Theorems 7-8 modified the existing Theorems 5-6 of [5].