Certain Fractional Proportional Integral Inequalities via Convex Functions

: The goal of this article is to establish some fractional proportional integral inequalities for convex functions by employing proportional fractional integral operators. In addition, we establish some classical integral inequalities as the special cases of our main ﬁndings.


Introduction
Integral inequalities play a vital role in the field of fractional differential equations. In the past few decades, researchers have paid their valuable consideration to this area. The significant developments in this area have been investigated, for example, [1][2][3], and [4] (cf. references cited therein). In [5], Ngo et al. established the following inequalities where σ > 0 and the positive continuous function g on [0, 1] such that Later on, Liu et al. [6] established the following inequalities where σ > 0, γ > 0, and the positive continuous g on [a, b] is such that Liu et al. [7] derived two theorems for integral inequalities as follows: Suppose that the functions f 1 and g 1 are positive and continuous on [a, b], (a < b) with f 1 ≤ g 1 on [a, b] such that the function f 1 g 1 , (g 1 = 0) is decreasing and the function f 1 is increasing. Assume that the function Φ is a convex with Φ(0) = 0. Then, the following inequality holds

Theorem 2.
Suppose that the functions f 1 , f 2 , and f 3 be positive and continuous is decreasing and the functions f 1 and f 3 are increasing. Assume that the function Φ is a convex with Φ(0) = 0. Then, the following inequality holds The inequalities in Equations (1)-(3) and their various generalizations have gained attention of the researchers [8][9][10][11][12].
Furthermore, the research of fractional integral inequalities is also of prominent importance. In [13,14], the authors presented some weighted Grüss type and new inequalities involving Riemann-Liouville (R-L) fractional integrals. In [15], Nisar et al. introduced many inequalities for extended gamma and confluent hypergeometric k-functions. Certain Gronwall inequalities for R-L and Hadamard k-fractional derivatives with applications are observed in [16]. The inequalities concerning the generalized (k, ρ)-fractional integral operators can be seen in [17].
The generalized fractional integral and Grüss type inequalities via generalized fractional integrals can be found in [18,19]. In [20], the authors examined the (k, s)-R-L fractional integral and its applications. In [21], the authors presented generalized Hermite-Hadamard type inequalities through fractional integral operators. Dahmani [22] introduced some classes of fractional integral inequalities by employing a family of n positive functions. Further the applications of fractional integral inequalities can be found [23,24].
In the last few decades, the researchers have paid their valuable consideration to the field of fractional calculus. This field has received more attention from various researchers due to its wide applications in various fields. In the growth of fractional calculus, researchers concentrate to develop several fractional integral operators and their applications in distinct fields (see, e.g., [25][26][27][28][29][30][31][32][33]). Zaher et al. [34] presented a new fractional nonlocal model.
Such types of these new fractional integral operators promote the future study to develop certain new approaches to unify the fractional operators and secure fractional integral inequalities. Especially, several striking inequalities, properties, and applicability for the fractional conformable integrals and derivatives are recently studied by various researchers. We refer the interesting readers to the works by [35][36][37][38][39][40][41][42][43][44], and [45]. The applications of conformable derivative can be found in [46][47][48][49] (cf. references cited therein).

Preliminaries
Jarad et al. [50] proposed the following left and right generalized proportional integral operators, which are sequentially defined by It follows that Multiplying Equation (7) by g(ρ)g(θ), we have Multiplying Equation (8) by 1 Then, it follows that Again, multiplying both sides of Equation (9) by 1 and integrating the resultant inequality with respect to θ over [a, τ], a < τ ≤ b, we get It follows that .
Now, since f ≤ g on [a, b] and Φ(τ) τ is an increasing function, for ρ ∈ [a, τ], a < τ ≤ b, we have Multiplying both sides of Equation (11) by 1 integrating the resultant inequality with respect to ρ over [a, τ], a < τ ≤ b, we get which, in view of Equation (4), can be written as Hence, from Equations (10) and (12), we get Equation (6).