Trapezium-Type Inequalities for k-Fractional Integral via New Exponential-Type Convexity and Their Applications

Department of Mathematics, Faculty of Technical Science, University Ismail Qemali, Vlora, Albania Department of Mathematics, Riphah International University, Faisalabad Campus, Satyana Road, Faisalabad, Pakistan Comsats University Islamabad, Lahore Campus, Lahore, Pakistan Virtual University of Pakistan, Lahore Campus, Lahore, Pakistan Department of Mathematics, College of Arts and Sciences, Prince Sattam Bin Abdulaziz University, Wadi Aldawaser 11991, Saudi Arabia Department of Mathematics and General Sciences, Prince Sultan University, Riyadh 12345, Saudi Arabia Department of Medical Research, China Medical University, Taichung 40402, Taiwan Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan


Introduction
eory of convexity also played a significant role in the development of theory of inequalities. Many famously known results in inequalities theory can be obtained using the convexity property of the functions. Hermite-Hadamard's double inequality is one of the most intensively studied results involving convex functions. is result provides us necessary and sufficient condition for a function to be convex. It is also known as classical equation of H-H inequality.
e Hermite-Hadamard inequality assert that if a function ψ: J ⊂ R ⟶ R is convex in J for a 1 , a 2 ∈ J and a 1 < a 2 , then Interested readers can refer to .
Definition 1 (see [30]). A function ψ: [0, +∞) ⟶ R is said to be s-convex in the second sense for a real number s ∈ (0, 1] or ψ belongs to the class of K 2 s , if ψ χa 1 holds for all a 1 , a 2 ∈ [0, +∞) and χ ∈ [0, 1]. A s-convex function was introduced in Breckner's article in [30], and a number of properties and connections with s-convexity in the first sense are discussed in [10]. Usually, convexity means for s-convexity when s � 1. Dragomir et al. proved a variant of Hadamard's inequality in [5], which holds for s-convex functions in the second sense. In the last decade, many mathematicians added the rich literature in the field of mathematical inequalities involving fractional calculus (see [8,9,13,15,22,25,28,31]).
Toader introduced the class of m-convex functions in [26].
Let ψ ∈ L[a 1 , a 2 ]. en, k− fractional integrals of order α, k > 0 with a 1 ≥ 0 are defined as follows: where Γ k (α) is the k− gamma function defined as We can notice that By choosing k � 1, the above k− fractional integrals yield Riemann-Liouville integrals.
Also that, the incomplete gamma function c(ϑ, θ) is defined for ϑ > 0 and θ ≥ 0 by integral e gamma function Γ(ϑ) is defined for ϑ > 0 by integral Motivated by above results and literatures, we will give first in Section 2 the concept of (s, m)-exponential-type convex function, and we will study some of their algebraic properties. In Section 3, we will prove new generalizations of Hermite-Hadamard-type inequality for the (s, m)-exponential-type convex function ψ and for the products of two (s, m)-exponential-type convex functions ψ and ϕ. In Section 4, we will obtain some refinements of the (H-H) inequality for functions whose first derivative in absolute value at certain power is (s, m)-exponential-type convex. In Section 5, some new bounds for special means and error estimates for the trapezoidal and midpoint formula will be provided. In Section 6, a briefly conclusion will be given as well.

Some Algebraic Properties of (s, m)-Exponential-Type Convex Functions
In this section, we will introduce a new definition, called (s, m)-exponential-type convex function, and we will study some basic algebraic properties of it.

New Generalizations of (H-H)-Type Inequality
In this section, we will establish some new generalizations of Hermite-Hadamard-type inequality for the (s, m)-exponential-type convex function ψ and for the products of two (s, m)-exponential-type convex functions ψ and ϕ.
Proof. Let denote, respectively, Using (s, m)-exponential-type convexity of ψ, we have Now, integrating on both sides in the last inequality with respect to χ over [0, 1], we get ψ a 1 + ma 2 2 which completes the left-side inequality. For the right-side inequality, using (s, m)-exponential-type convexity of ψ, we obtain which gives the right-side inequality. □ Corollary 1. By taking m � s � 1 in eorem 5, we get ( eorem 5, [11]). Theorem 6. Assume that ψ, ϕ: [a 1 , ma 2 ] ⟶ R are, respectively, (s 1 , m) and (s 2 , m)-exponential-type convex functions for the same fixed m ∈ (0, 1] and for some fixed s 1 , s 2 ∈ (0, 1], where s 1 < s 2 and a 1 < ma 2 . If ψ, ϕ are synchronous functions and ψ, ϕ, ψϕ ∈ L 1 ([a 1 , ma 2 ]), then where Using the property of the (s 1 , m) and (s 2 , m)-exponentialtype convex functions ψ and ϕ, respectively, we have Multiplying above inequalities on both sides, we get Applying Chebyshev integral inequality (see [23]), we obtain Changing the variable of integration, we get which completes the left-side inequality. For the right-side inequality, integrating on both sides of the inequality (25) with respect to χ over [0, 1], we have which give the right-side inequality. □

Refinements of (H-H)-Type Inequality for k − Fractional Integral
In this section, we will obtain some refinements of the (H-H) inequality via (s, m)-exponential-type convex functions.
Proof. Using the integrating by parts, we have which completes the proof.

Applications
Let consider the following two special means for different positive-real numbers a 1 < a 2 .
Using Section 4, we have the following interesting results: en, for some fixed s ∈ [ln2.5, 1), where Proof. Consider the s-exponential convex function ψ(x) � x ls , x > 0, and using eorem 9 for α � 1 � k, the result is evident. □ Proposition 5. Let 0 < a 1 < a 2 , 0 < w ≤ 1 and q ≥ 1. en, for some fixed s ∈ [ln2.5, 1), where 1 ≤ l ≤ 1/s, we have Proof. Consider the s-exponential convex function ψ(x) � x ls , x > 0, and using eorem 10 for α � 1 � k, we obtain the required result. At the end, let consider some applications of the integral inequalities obtained above, to find new bounds for the trapezoidal and midpoint formula.

Conclusion
In this article, the authors showed new generalizations of trapezium-type inequality for the new class of functions, the so-called (s, m)-exponential-type convex function ψ and for the products of two (s, m)-exponential-type convex functions ψ and ϕ. We have obtained refinements of the (H-H) inequality for functions using (s, m)-exponential-type convex and founded new bounds for special means and for the error estimates for the trapezoidal and midpoint formula. We hope that current work will attract the attention of researchers working in mathematical analysis, fractional calculus, quantum calculus, postquantum calculus, and other related fields.

Data Availability
No data were used to support this study.

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