New Developments of Hermite–Hadamard Type Inequalities via s -Convexity and Fractional Integrals

. In this paper, we present an identity for diferentiable functions that has played an important role in proving Hermite–Hadamard type inequalities for functions whose absolute values of frst derivatives are s -convex functions. Meanwhile, some Hermite–Hadamard type inequalities for the functions whose absolute values of second derivatives are s -convex are also established with the help of an existing identity in literature. Many limiting results are deduced from the main results which are stated in remarks. Some applications of proved results are also discussed in the present study.


Introduction
Inequalities have been proved to be the most efcient tools for the construction of several branches in mathematics.In the feld of classical diferential and integral equations, the inequalities have played an important role [1,2].Charles Hermite and Jacques Hadamard derived Hermite-Hadamard inequality which is stated as follows.
Theorem 1 (see [23]).Let ϕ: [ω, ]] ⟶ R be a positive function with 0 ≤ ω ≤ ] and ϕ ∈ L[ω, ]].If ϕ is a convex function on [ω, ]], then the following inequality for fractional integrals holds: where J θ ω + ϕ and J θ ] − ϕ indicate the left-sided and right-sided Riemann-Liouville fractional integrals of the order θ ∈ R + � [0, ∞) which are as follows: respectively, and Γ(•) is the classical Euler gamma function.s-convexfunctions are generalization of classical convex function.It is remarkable that Özdemir et al. [24] defned the s-convex function as follows: s − convex function [24].Let ϕ: R ⟶ R be a function, then ϕ is called s-convex function, if for each ω, ] ∈ R and c ∈ (0, 1), s ∈ (0, 1].Hölder Inequality for Integrals [25].Let p > 1 and 1/p + 1/p � 1.If ϕ and ξ are real functions on [ω, ]] and if |ϕ| p , |ξ| q are integrable functions on [ω, ]], q ≥ 1, then Te following power-mean integral inequality is an elementary result of Hölder inequality: Power-Mean Integral Inequality [25].Let q ≥ 1.If ϕ and ξ are real functions defned on Some authors applied classical inequalities such as Hölder inequality and power mean inequality and also applied the special functions like classical Euler-gamma and beta functions to fractional integrals to get new integral inequalities for the diferent classes of convex functions.Qaisar et al. obtained some new Hermite-Hadamard inequalities involving fractional integrals for convex functions [14].Some refnements for integral and sum forms of Hölder inequality were elaborated by Is ¸can [1].Authors are motivated by the results given in [3,26].Te purpose of this paper is to establish new Hermite-Hadamard type inequalities involving fractional integrals via s-convex functions.

Hermite-Hadamard Type Inequalities Involving Fractional Integrals for the Class of Differentiable Functions
To prove our main results associated with Hermite-Hadamard type inequalities involving fractional integrals, we need the following lemma.

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Substituting l � cr + (1 − c)ω in the above equation, we get Now, consider Substituting I � cr + (1 − c)] in the above equation, we get Adding equations ( 9) and ( 11), we get Te proof is completed.

□
Remark 3. By replacing r with ] in equation ( 9) and r with ω in equation (11) and adding the resulting equations, we obtain the following equation: Substituting c � 1 − l in the second term of R. H. S of the equation (13), then equation (13) It is easy to see that ξ ″ (c) ≥ 0 for 0 ≤ c ≤ 1 and ξ(1) � ξ(0) � 0, so ξ(c) ≤ 0, therefore and that means Journal of Mathematics Example 2. Let ϕ: (0, ∞] ⟶ R be defned by ϕ(r) � r − r ln r, 0 < s ≤ 1, and 0 so ξ(t) ≥ 0, therefore Theorem 4. Let θ ≥ 1 and ϕ: , then the following inequality for fractional integrals holds: Proof.Applying the s-convexity of ϕ, we get Multiply both sides of equation ( 20) by Now, substituting l � cr + (1 − c)ω in the frst term of L. H. S of equation ( 21) and I � cr + (1 − c)] in the second term of L. H. S of equation ( 21), we get Multiplying both sides of the above equation by θ, we get Te proof is completed.
then the following inequality for fractional integral holds: for some fxed s Te proof is completed.
then the following inequality for fractional integral holds: for some fxed s ∈ (0, 1].
Proof.According to Lemma 2,

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Applying the Hölder inequality (5) in equation ( 28), we get Te proof is completed.

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According to Hölder inequality (5), we get Te proof is completed.

□ . Hermite-Hadamard Type Inequalities Involving Fractional Integrals for the Class of Twice Differentiable Functions
Dragomir et al. [27] defned the following identity involving Riemann-Liouville fractional integrals.

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Proof.According to Lemma 9, we get According to Hölder inequality (5),
Proof.According to Lemma 9, we get

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Applying the Hölder inequality (5), we get Te proof is competed.

Applications to Some Special Means
Consider the following special means for arbitrary real numbers f, g and f ≠ g as follows: ⎤ ⎦ . (72) Proof.Tis statement follows from Teorem 13, by using ϕ(c) � 1/c, c ≠ 0, and θ � 1.

Conclusion
In this paper, authors have established Hermite-Hadamard type inequalities involving Riemann-Liouville fractional integrals via s-convex functions by applying two diferent techniques.In frst part, an identity is proved in which a diferentiable function is presented in the form of Riemann-Liouville fractional integrals of frst derivatives of function.Furthermore, this identity is used to establish Hermite-Hadamard type inequalities in which the absolute values of frst derivatives are s-convex functions.In the second part, an identity in which a function in the form of integral of double derivative of function is used to establish Hermite-Hadamard inequalities in which the absolute values of second derivatives are s-convex functions.Te limiting cases included some existing results in the literature.Some applications of the obtained results are also described in the form of means.Tis method can also be applicable for other classes of convex functions.