Some New Refinements of Hermite–Hadamard-Type Inequalities Involving ψk-Riemann–Liouville Fractional Integrals and Applications

,e above inequality is known as Hermite–Hadamard’s inequality [1–5]. ,is inequality provides us a necessary and sufficient condition for a function to be convex. It can be considered as one of the most extensively studied results pertaining to convexity. Since the appearance of this result in the literature, it gained popularity, and many new generalizations for this classical result have been obtained. ,is can be attributed to its applications in various other fields such as in numerical analysis and in mathematical statistics. For more details on generalizations of convexity, Hermite–Hadamard-like inequalities, and its applications, see [6–14]. Fractional calculus is a calculus in which we study about the integrals and derivatives of any arbitrary real or complex order.,e history of fractional calculus is not very much old, but in the short span of time, it experienced a rapid development. Recently, the generalizations [15–25], extensions [26–32], and applications [33–46] for fractional calculus have been made by many researchers. ,e Riemann–Liouville fractional integrals are defined as follows.


Introduction and Preliminaries
Let f: I � [a, b] ⊂ R ⟶ R be a convex function; then, e above inequality is known as Hermite-Hadamard's inequality [1][2][3][4][5]. is inequality provides us a necessary and sufficient condition for a function to be convex. It can be considered as one of the most extensively studied results pertaining to convexity. Since the appearance of this result in the literature, it gained popularity, and many new generalizations for this classical result have been obtained. is can be attributed to its applications in various other fields such as in numerical analysis and in mathematical statistics. For more details on generalizations of convexity, Hermite-Hadamard-like inequalities, and its applications, see [6][7][8][9][10][11][12][13][14].
e Riemann-Liouville fractional integrals are defined as follows.
en, Riemann-Liouville integrals J α a + f and J α b − f of order α > 0 with a ≥ 0 are defined by where Sarikaya et al. [10] elegantly utilized this concept in establishing fractional analogue of Hermite-Hadamard's inequality.
is idea motivated other researchers, and consequently, many new generalizations of Hermite-Hadamard's inequality have been obtained using the concept of Riemann-Liouville fractional integrals.
Sarikaya and Karaca [12] introduced k-analogue of Riemann-Liouville fractional integrals and discussed some of its basic properties. ey defined this concept in the following way: to be more precise, let f be piecewise continuous on I * � (0, ∞) and integrable on any finite sub- en, for t > 0, we consider k-Riemann-Liouville fractional integral of f of order α as If k ⟶ 1, then k-Riemann-Liouville fractional integrals reduce to classical the Riemann-Liouville fractional integral. It is worth to mention here that the concept of the k-Riemann-Liouville fractional integral is a significant generalization of Riemann-Liouville fractional integrals; as for k ≠ 1, the properties of k-Riemann-Liouville fractional integrals are quite different from the classical Riemann-Liouville fractional integrals.
Another important generalization of Riemann-Liouville fractional integrals is ψ k -Riemann-Liouville fractional integrals.
Definition 2 (see [6]). Let (a, b) be a finite interval of the real line R and α > 0. Also, let ψ(x) be an increasing and positive monotone function on (a, b], having a continuous derivative ψ ′ (x) on (a, b). en, the left-and right-sided ψ-Riemann-Liouville fractional integrals of a function f with respect to another function ψ on [a, b] are defined as respectively; Γ(·) is the gamma function.
For some recent research works, see [48].
Recently, Liu et al. [14] obtained some interesting results pertaining to Hermite-Hadamard's inequality involving ψ k -Riemann-Liouville fractional integrals. Motivated by the research work of Liu et al. [14], we obtain some new refinements of fractional Hermite-Hadamard's inequality essentially using ψ k -Riemann-Liouville fractional integrals. We also discuss applications of the obtained results to means. We show that our results represent significant generalization of some previous results.

Hermite-Hadamard's Inequality
In this section, we derive a new refinement of Hermite-Hadamard's inequality via the ψ k -Riemann-Liouville fractional integral. en, the left-and right-sided ψ k -Riemann-Liouville fractional integrals of a function f with respect to another function ψ on [a, b] are defined as respectively; is the k-analogue of gamma function. e k-analogues of beta function and incomplete beta function are, respectively, defined as We now derive the main result of this section. on (e, f), and α ∈ (0, 1). en, for k > 0, the following k-fractional integral inequalities hold: Proof. Using the convexity of g, we have 2g Multiplying both sides by t (α/k)− 1 and then integrating with respect to t on [0, 1], we have Now, making the substitution t � (ψ Mathematical Problems in Engineering Also, using the convexity property of g, we have Multiplying both sides by t (α/k)− 1 and then integrating it with respect to t on [0, 1], we obtain e proof is completed.

Some More Fractional Inequalities of Hermite-Hadamard Type
We now derive two new fractional integral identities involving ψ k -Riemann-Liouville fractional integrals. ese results will serve as auxiliary results for obtaining our next results. on (e, f), and α ∈ (0, 1). en, for k > 0, the following identity holds: Proof. Consider Similarly,

Mathematical Problems in Engineering
It follows that en, all the assumptions in Lemma 1 are satisfied. Observe that (g(c) + g(d)/2) � (13/2).
is implies Also, where h � Proof. Suppose Summing I 1 , I 2 , I 3 , and I 4 , we get the required result.
Mathematical Problems in Engineering 5 Also, where h is defined in Lemma 2.
is implies Also, where h is defined in Lemma 2.
is implies Before proceeding to next results, let us recall the definition of s-convex function of Breckner type.
Proof. Using Lemma 1 and the fact that |g ′ | is Breckner type of s-convex function, we have 6 Mathematical Problems in Engineering where is completes the proof. on (e, f) and α ∈ (0, 1), then for k > 0, the following inequality holds: where L 1 and L 2 are given by (44) and (45), respectively.
Proof. Using Lemma 2, the property of modulus, and the given hypothesis of the theorem, we have Mathematical Problems in Engineering 7 Using substitution t � (ψ(v) − e/f − e) and the fact that |g ′ | is Breckner type of s-convex function, we have where L 1 and L 2 are given by (44) and (45), respectively. And is completes the proof.

Applications
In this section, we discuss some applications of eorem 2 to means by considering a particular example of s-convexity.

(55)
We now give the main results of this section. where respectively.

Conclusion
In this article, we obtain some new fractional estimates of Hermite-Hadamard's inequality essentially using a new k-analogue of ψ k -fractional integrals. We derive two new fractional integral identities in the setting of k-fractional calculus. In order to check the validity of these identities, we discuss some particular examples. In the final section, we have discussed applications of eorems 2 and 3 to means.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.