Caputo Fractional Derivative Hadamard Inequalities for Strongly m-Convex Functions

In this paper, two versions of the Hadamard inequality are obtained by using Caputo fractional derivatives and strongly m-convex functions. The established results will provide refinements of well-known Caputo fractional derivative Hadamard inequalities form -convex and convex functions. Also, error estimations of Caputo fractional derivative Hadamard inequalities are proved and show that these are better than error estimations already existing in literature.


Introduction
Strongly convex function was introduced by Polyak in [1]. Strong convexity is a strengthening of the notion of convexity; some properties of strongly convex functions are just "stronger versions" of known properties of convex functions. Strongly convex functions have been used for proving the convergence of a gradient-type algorithm for minimizing a function. They play an important role in the optimization theory and mathematical economics. Definition 1. Let D be a convex subset of X, ðX, k:kÞ be a normed space. A function ψ : D ⊂ X ⟶ R is called strongly convex function with modulus C if it satisfies ∀a, b ∈ D, z ∈ ½0, 1 and C ≥ 0. Many authors have been inventing the properties and applications of strongly convex function, for detailed information, see [2][3][4][5][6].
The concepts of m-convex functions and strongly mconvex functions were introduced in [7,8], respectively. Toader [7] gave the idea of m-convex functions as follows.
Definition 2. A function ψ : ½0, b ⟶ ℝ is said to be mconvex, where m ∈ ½0, 1, if for every x, y ∈ ½0, b and z ∈ ½0, 1, we have Lara et al. introduced strongly m-convex functions as follows: Definition 3 (see [8]). A function ψ : I ⟶ ℝ is called strongly m-convex function with modulus C ≥ 0 if for a, b ∈ I and z ∈ ½0, 1: Nowadays, fractional integral inequalities are in the study of several researchers (see, [9][10][11][12][13][14][15][16][17] and references therein), where they have used different kinds of well-known functions and fractional integral operators. Here, in this paper, we are interested to produce fractional integral inequalities for Caputo fractional derivatives of strongly m-convex functions. The Caputo fractional derivative operators are defined as follows.
The Hadamard inequality is another interpretation of convex function. It is stated as follows.
Definition 5 (see [19]). Let ψ : I ⟶ ℝ be a convex function on interval I ⊂ ℝ and a, b ∈ I where a < b. Then, the following inequality holds: If order in (6) is reversed, then it holds for concave function.
Farid et al. [20] have proved the following Hadamard inequality for Caputo fractional derivatives of convex functions: Theorem 6. Let ψ : ½a, b ⟶ ℝ be the function with ψ ∈ AC n ½a, b and 0 ≤ a < b. Also, let ψ ðnÞ be positive and convex function on ½a, b: Then, the following inequality holds for Caputo fractional derivatives: They also established the following identity.
be the function with ψ ∈ AC n+1 ½a, b and also let jψ ðn+1Þ j be convex on ½a, b: Then, the following inequality for Caputo fractional derivatives holds: Kang et al. [21] proved the following version of the Hadamard inequality for Caputo fractional derivatives. Theorem 9. Let ψ : ½a, b ⟶ ℝ be a positive function with ψ ∈ AC n ½a, b and 0 ≤ a < b. If ψ ðnÞ is convex function on ½a, b, then the following inequality for Caputo fractional derivatives holds: Farid et al. [22] established the following identity.
Journal of Function Spaces the following equality for Caputo fractional derivatives holds: Kang et al. [21] also proved the following inequalities for Caputo fractional derivatives.
Theorem 11. Let ψ : ½a, b ⟶ ℝ be a differentiable mapping on ða, bÞ with ψ ∈ AC n+1 ½a, b and a < b. If jψ ðn+1Þ j q is convex on ½a, b for q ≥ 1, then the following inequality for Caputo fractional derivatives holds: Theorem 12 (see [21]). Let ψ : ½a, b ⟶ ℝ be a function with ψ ∈ AC n+1 ½a, b and a < b. If jψ ðn+1Þ j q is convex on ½a, b for q > 1, then the following inequality for Caputo fractional derivatives holds: We will study all of the above fractional inequalities for strongly m-convex functions and at the same time will obtain their generalizations and refinements. In Section 2, we will give refinements of two versions of the Hadamard inequality for Caputo fractional derivatives. We will connect their particular cases with some well-known results. In Section 3, by applying known identities, we will give refinements of error estimations of the Hadamard inequalities.

Main Results
The following result is the generalization of Theorem 6 which in a particular case also provides its refinement.

Journal of Function Spaces
By multiplying (16) with z n−β−1 on both sides and making integration over ½0, 1, we get By using change of variables and computing the last integral, from (17), we get Further, it takes the following form Since ψ ðnÞ is strongly m-convex function with modulus C, for z ∈ ½0, 1, then one has By multiplying (20) with z n−β−1 on both sides and making integration over ½0, 1, we get By using change of variables and computing the last integral, from (21), we get Further, it takes the following form Inequalities (19) and (11) constituted the required inequality.
The consequences of Theorem 13 are stated in the following corollary and remark: 4 Journal of Function Spaces Corollary 14. By setting m = 1 in inequality (14), we will get ( [23], Theorem 6) Remark 15. If C = 0 and m = 1 in (14), then we will get the fractional Hadamard inequality stated in Theorem 6. The upcoming result is the refinement of another version of the Hadamard inequality for Caputo fractional derivatives stated in a theorem in [19].

Theorem 16.
Under the assumptions of Theorem 13, the following inequality for Caputo fractional derivatives holds: with β > 0.

Journal of Function Spaces
By multiplying (30) with z n−β−1 on both sides and making integration over ½0, 1, we get By using change of variables and computing the last integral, from (31), we get Further, it takes the following form From (29) Remark 18. If C = 0 and m = 1 in (25), then we will get the fractional Hadamard inequality stated in Theorem 9.

Error Bounds of Fractional Hadamard Inequalities
In this section, we give refinements of the error bounds of fractional Hadamard inequalities for Caputo fractional derivatives.
Proof. Since |ψ ′ | is strongly m-convex function on ½a, b, for z ∈ ½0, 1, we have Journal of Function Spaces By applying Lemma 7 and the strongly m-convexity of j ψ ðn+1Þ j, we find In the following, we compute integrals appearing on the right side of inequality (37):