Some New Kinds of Fractional Integral Inequalities via Refined (α, h−m)-Convex Function

Convex functions are important in diverse fields of mathematics, statistics, engineering, and optimization. Especially in the formation of inequalities, they play a very vital role. In the subject of mathematical analysis, inequalities provide a significant contribution in developing classical concepts and notions. For example, inequalities well known as Cauchy–Schwarz inequality, Chebyshev inequality, Minkowski inequality, Hadamard inequality, and Jensen inequality are utilized frequently in pure and applied mathematics. It is always a challenge to extend, generalize, and refine such inequalities by considering new classes of functions. In this era, researchers are working on classical inequalities concerning fractional integral and derivative operators. It can be observed that the Hadamard inequality is studied more for many kinds of fractional integral and derivative operators than any other classical inequality, see [1–7] for more details. (e aim of this paper is to study the refinements of Hadamard and other integral inequalities recently studied in [8–11]. (e consequences of these inequalities also provide refinements of fractional integral inequalities connected with the integral inequalities studied in the recent past. (e article is organized as follows. In Section 2, we suggest some preliminaries. In Section 3, the bounds of unified integral operators are given using refined (α, h − m)-convex functions. (ese are the refinements of bounds already obtained in the literature. In Section 4, some applications of the main results are given in the form of fractional integral inequalities and their refinements.


Introduction
Convex functions are important in diverse fields of mathematics, statistics, engineering, and optimization. Especially in the formation of inequalities, they play a very vital role. In the subject of mathematical analysis, inequalities provide a significant contribution in developing classical concepts and notions. For example, inequalities well known as Cauchy-Schwarz inequality, Chebyshev inequality, Minkowski inequality, Hadamard inequality, and Jensen inequality are utilized frequently in pure and applied mathematics. It is always a challenge to extend, generalize, and refine such inequalities by considering new classes of functions. In this era, researchers are working on classical inequalities concerning fractional integral and derivative operators. It can be observed that the Hadamard inequality is studied more for many kinds of fractional integral and derivative operators than any other classical inequality, see [1][2][3][4][5][6][7] for more details. e aim of this paper is to study the refinements of Hadamard and other integral inequalities recently studied in [8][9][10][11]. e consequences of these inequalities also provide refinements of fractional integral inequalities connected with the integral inequalities studied in the recent past. e article is organized as follows. In Section 2, we suggest some preliminaries. In Section 3, the bounds of unified integral operators are given using refined (α, h − m)-convex functions. ese are the refinements of bounds already obtained in the literature. In Section 4, some applications of the main results are given in the form of fractional integral inequalities and their refinements.
Inequality (6) gives refinements of several types of convexities when 0 < h(t) < 1, see [14]. e need for integral operators in the study of fractional derivatives is of immense importance. In the recent era, integral operators are being used extensively for producing new results in the literature. For references, see [2,[4][5][6]. Next, we give some fundamental integral operators which are used in this paper.
e left and right fractional integrals of Ω with respect to Δ on [x 1 ′ , y 1 ′ ] of order κ are given by where Γ(.) is the gamma function and R(κ) > 0.

Main Results
roughout the paper, we use the following notation: Theorem 1. Let Ω be a positive, refined (α, h − m)-convex and integrable function defined over [x 1 ′ , y 1 ′ ]. Also, let Υ/x be an increasing function defined on [x 1 ′ , y 1 ′ ] and Δ be strictly increasing and differentiable function on (x 1 ′ , y 1 ′ ). en, for β, ξ, c, ι ∈ R, p, κ, ϑ, δ ≥ 0, 0 < k ≤ δ + κ, and 0 < k ≤ δ + ϑ, the following result holds: Proof. For the functions Υ/x and Δ, the following inequality holds: Using refined (α, h − m)-convexity of Ω, one can have From (17) and (18), we have the following integral inequality: Mathematical Problems in Engineering Using (12) of Definition 10 on the left side of inequality (19) and making change of the variable by setting u � x − t/x − x 1 ′ on the right-hand side of the above inequality, we obtain Also, for t ∈ (x, y 1 ′ ] and x ∈ (x 1 ′ , y 1 ′ ), we can write and From (22) and (23), we have the following integral inequality: Using (13) of Definition 10 on the left-hand side and making change of the variable by setting v � t − x/y 1 ′ − x on the right-hand side of the above inequality, we obtain Mathematical Problems in Engineering erefore, Combining (21) and (26), the required inequality (16) is obtained. Hence, the proof is completed.
Next, we give the refinement of eorem 1.
Now, we give the refinement of eorem 5 in [9] in the following corollary.
Next, we give a lemma which we will use in the proof of upcoming eorem 3.

Mathematical Problems in Engineering
Using (13) of Definition 10 on the right-hand side and making change of the variable by setting v � x − x 1 ′ /y 1 ′ − x 1 ′ on the right-hand side of the above inequality, we obtain e following inequality also holds true for x ∈ (x 1 ′ , y 1 ′ ): From (35) and (38), the following integral inequality is obtained: (39) Using (12) of Definition 10 on the left-hand side and making change of the variable on the right-hand side of the above inequality, we obtain Now, using Lemma 1, we can write which by using (13) of Definition 10 gives the following integral inequality: Again, using Lemma 1, we can write which by using (12) of Definition 10 gives the following fractional integral inequality: Proof. From (35), one can see that, for 0 < h(t) < 1, Hence, by following the proof of eorem 3, one can obtain (45). is completes the proof.

Mathematical Problems in Engineering
Now, we give the refinement of eorem 6 in [9] in the following corollary.

Remark 3
(i) For h(t) � t and m � 1 � α, inequality (33) coincides with eorem 5 in [19] (ii) For h(t) � t and m � 1 � α, inequality (47) coincides with Corollary 2 in [19] By using 0 < h(t) < 1 and making different choices of functions h and Δ and the parameters in (33), one can get the refinements of many well-known inequalities for different classes of convex functions which are mentioned in Remark 5 of [9].

Corollary 10.
For k � 1 in Corollary 9, the following upper bound for Riemann-Liouville fractional integral is satisfied: Remark 7. For 0 < h(t) < 1, (69) gives refinement of Corollary 11 in [9]. Similar bounds can be obtained for eorems 3 and 5, which we leave for the reader.

Conclusions
is article is about the bounds of unified integral operators via refined (α, h − m)-convexity. e obtained results are the refinements of some already published results. Moreover, some deducible fractional integral operators and their related bounds are also given.

Data Availability
No data were used to support this study.

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