Riemann-Liouville Fractional Versions of Hadamard inequality for Strongly m-Convex Functions

This paper deals with Hadamard inequalities for strongly m-convex functions via RiemannLiouville fractional integrals. These inequalities provide refinements of well known fractional integral inequalities for convex functions. Further, by applying an identity error estimations are obtained and compared with already known error estimations.


Introduction
First, we will give some definitions and well known results which are needful and connected with the findings of this paper. holds ∀ x = y ∈ I and t ∈ (0, 1).
Definition 1.2. [7] Let (X, ||.||) be a normed space. A function f : E ⊂ X → R will be called strongly convex function with modulus C if holds ∀ x, y ∈ E ⊆ X, t ∈ [0, 1]. The Hadamard inequality is another way of representing convex function stated in the upcoming theorem.
[2] Let f : I → R be a convex function on interval I ⊂ R and x, y ∈ I where x < y .
Then the following inequality holds: If order in inequality (1.6) is reversed, then it holds for concave function.
The Riemann-Liouville fractional integrals are defined as follows: The Riemann-Liouville fractional integral operators of order β > 0 are defined as follows: (1.8) The following version of the Hadamard inequality for convex functions via Riemann-Liouville fractional integrals was proved by Sarikaya et al. in [9]: with β > 0.
In [9], they also studied the error estimations of this fractional Hadamard inequality by establishing an identity. Another version of the Hadamard inequality was proved by Sarikaya and Yildirim in [11].
, then the following fractional integrals equality holds: (1.10) Our aim in this paper is to study all of the above inequalities and their error estimations by applying definition of strongly m-convex functions.
In the upcoming section we will prove the version of Hadamard inequality for strongly m-convex functions, which simultaneously will represent refinement as well as generalization of Theorem 1.2. Another version of the Hadamard inequality will be proved which will provide refinement and generalization of Theorem 1.3 at the same time. Also error estimations of the fractional Hadamard inequality are given in refined form.
Since the function f is strongly m-convex function, for u, v ∈ [x, y ] we have By multiplying inequality (2.3) with t β−1 on both sides and then integrating over the interval [0, 1], By change of variables we will get .
Therefore, the above inequality takes the following form: .
From the definition of strongly m-convex function with modulus C, for t ∈ [0, 1] we have the following inequality: By multiplying inequality (2.7) with t β−1 on both sides and then integrating over the interval [0, 1], we get By change of variables we will get .
Therefore, the above inequality takes the following form: .
(iii) For m = 1 in inequality (2.1) we get the Hadamard inequality for strongly convex function.
(iv) For β = 1 and m = 1 in (2.1) we will get the refinement of the Hadamard inequality.
Proof. For t ∈ [0, 1] and strongly m-convexity of function, let u = x t 2 +m( 2−t 2 )y and v = ( 2−t 2 ) x m +y t 2 in inequality (2.2), we have By multiplying (2.12) with t β−1 on both sides and making integration over [0, 1] we get By using change of variables and computing the last integral, from (2.13) we get Further it takes the following form .
The first inequality of (2.11) can be seen in (2.15). Now we prove the second inequality of (2.11).
Since f is strongly m-convex function and t ∈ [0, 1], we have the following inequality: By using change of variables and computing the last integral, from (2.17) we get .
Further it takes the following form . (ii) For β = 1 and m = 1 in (2.1) we will get refinement of the Hadamard inequality.
(iv) For m = 1 in inequality (2.11) we will get the Hadamard inequality for strongly convex function.

Error Estimations
then the following fractional integrals equality holds: inequality holds: with β, C > 0.
Proof. Since |f | is strongly m-convex function on [x, my ] and t ∈ [0, 1], we have

By using Lemma 3.1 and (3.3) we have
After simplify the last inequality of (3.4), we get the inequality (3.2).