Hermite-Hadamard Fractional Inequalities for Differentiable Functions

: In this article, we look at a variety of mean-type integral inequalities for a well-known Hilfer fractional derivative. We consider twice differentiable convex and s -convex functions for s ∈ ( 0,1 ] that have applications in optimization theory. In order to infer more interesting mean inequalities, some identities are also established. The consequences for Caputo fractional derivative are presented as special cases to our general conclusions.


Introduction
The subject of fractional calculus has achieved a significant prominence during the most recent couple of years due to its demonstrated applications in the field of science and engineering. This offers useful strategies to solve differential and integral equations, see the books [1,2] and articles [3][4][5]. Fractional calculus has been applied in different areas of science, engineering, financial mathematics, applied sciences, bio engineering, etc.
Mathematical inequalities are significantly important in the study of mathematics and related fields. Nowadays, fractional integral inequalities are fruitful in generating the uniqueness of solutions for fractional partial differential equations. They also provide boundedness of the solutions of fractional boundary value problems. These recommendations have inspired various researchers in the field of integral inequalities to inquire the extensions by involving fractional calculus operators. Recently, Peter Korus presented a class of Hermite-Hadamard inequalities by considering the class of convex or generalized convex derivative in [6], Farid et al. explored Fejér-Hadamard type inequalities [7] for (α, h − m) − p-convex functions by involving the fractional operators. We further refer the reader to, e.g., [8][9][10].
The convex functions are utilized to create numerous inequalities like Alomari et al. [11] present Ostrowski's inequalities via s convexity in second sense, Dragomir et al. discuss some properties of convex functions in [12] and explored some important quadrature rules in [13]. More applications can be observed from literature [14][15][16] on convex functions and inequalities. Hermite-Hadamard's inequality [17] is one of the most important classical inequalities, as it has a rich geometrical meaning and applications [18][19][20]. Hermite-Hadamard's double inequality is one of the most widely studied concerning convex functions. The inequality is defined as follows: Let ψ : I ⊆ R → R be a convex mapping and θ, ζ ∈ I with θ < ζ. Then, If ψ is concave, then the inequalities (1) hold in reverse direction. For particular choices of function ψ, some classical inequalities for means can be derived from (1) (see [21]). The principle point of this paper is to infer Hermite-Hadamard-type integral inequalities for Hilfer fractional derivative. Such inequalities were proved by many scientists for different convexities and for many fractional operators, but the main results of this paper are more general then the existing literature.

Preliminaries
In this section, we recall some basic preliminary results.
The definition of classical Riemann-Liouville fractional derivative (see [23] (Chapter 4)) is given as follows.
The more general integral representation of Equation (3) given in [24] is defined as follows: Let which coincide with (3) for n = 1. Specially for β = 0, D γ,0 θ + ψ = D γ θ + ψ is Riemann-Liouville fractional derivative of order γ and for β = 1 it is Caputo fractional derivative D γ,1 θ + ψ = C D γ θ + ψ of order γ. Applying the properties of Riemann-Liouville integral the relation (4) can be rewritten in the form The geometric arithmetically s-convex function given in [25] presented in the following definition.

Main Results
This section includes several mean-type fractional integral inequalities involving Hilfer fractional derivative. The first main result for the fractional derivative is presented in the following theorem.
ψ is convex function on [θ, ζ], then the following inequality for fractional derivative holds. .
The special case of Theorem 1 presented in [27] (Theorem 2.3) is given as follows.
The following special case of Lemma 3 was proved by Farid et al. in [27] (Lemma 2.2).
ψ is convex on [θ, ζ], then the following inequality is true

Proof. By using Lemma 3 and Definition 1, we get
Hence, the proof is complete.
The corollary given below presented in [27] (Theorem 2.4) is a special case of Theorem 2.
Corollary 3. If we choose β = 1 and ψ is symmetric about θ+ζ 2 in Theorem 2, we get , then we have the following equality.
Proof. By using Lemma 3, we get Integrating by parts, we get Since and By substituting again By adding (19) and (20), we obtain Using Equation (21) into (18), we get the required result.

Proof. Consider
Integrating I 1 by parts, we get Now integrating I 2 by parts, we get By substituting (23) and (24) to (22), we get By using Lemma (4), we arrive at the desired result.
Proof. By using Lemmas 1, 4 and Definition 4, we have By using the definition of the beta function, we get Which completes the proof of the result.
Corollary 6. If we take β = 1 and ψ is symmetric about θ+ζ 2 in Theorem 3, then the following result for Caputo fractional derivatives holds.