Some New Fractional Inequalities Involving Convex Functions and Generalized Fractional Integral Operator

In this manuscript, we are getting some novel inequalities for convex functions by a new generalized fractional integral operator setting. Our results are established by merging the ð k , s Þ -Riemann-Liouville fractional integral operator with the generalized Katugampola fractional integral operator. Certain special instances of our main results are considered. The detailed results extend and generalize some of the present results by applying some special values to the parameters.

There is one more appealing and useful inequality, namely, the Pólya-Szegö inequality [11], which establishes the essential key of motivation in our study, which we can indicate as follows where m ≤ f ðxÞ ≤ M and n ≤ gðxÞ ≤ N, for some m, M, n, N ∈ ℝ and for each x ∈ J ≔ ½a, b: In [12], Dragomir and Diamond introduced the following Grüss type integral inequality by the Pólya-Szegö result, here, 0 < m ≤ f ðxÞ ≤ M < ∞ and 0 < n ≤ gðxÞ ≤ N < ∞, for x ∈ J: The next integral inequalities have been proved by Ngo [13], where x > 0 and v are a positive continuous on ½0, 1 such that In this regard, Liu et al. [14] introduced the subsequent inequality where μ, ν > 0 and v are a positive continuous on J, and ς = min ð1, νÞ, for x ∈ J: Since one of the primary inspiration points of fractional analysis is getting more general and valuable fundamental integral operators, the generalized fractional integral operator is a decent tool to sum up numerous past investigations and results, see [15][16][17][18]. Essentially, in inequality theory, scientists utilize such broad operators to generalize and extend their inequalities.
Lately, these fractional integral operators have been considered and used to broaden particularly Grüss, Chebychev-Grüss, Pólya-Szegö, Gronwall, Minkowski, and Hermite-Hadamard type inequalities. For additional subtleties, Agarwal [19] proved some fractional integral inequalities with Hadamard's fractional integral operators. Ntouyas et al. [20] established certain Chebyshev type inequalities involving Hadamard's fractional integral operators. Some Grüss type inequalities under k-Riemann-Liouville fractional integral operators have been investigated by Set et al. [21]. In [22], the authors introduced some Pólya-Szegö type inequalities by Hadamard k-fractional integral operators. A new version for the Gronwall type inequality involving generalized proportional fractional integral operators was presented by Alzabut et al. [23]. Rahman et al. [24] established reverse Minkowski inequalities with generalized proportional fractional integral operators. Many Chebyshev type inequalities with generalized conformable fractional integral operators have been discussed by Nisar et al. [25].
In this regard, Dahmani [26] established some new inequalities for convex functions involving Riemann-Liouville fractional integral operator. Jleli et al. [27] obtained new Hermite-Hadamard type inequalities for convex functions via generalized fractional integral operators. Some Hermite-Hadamard type inequalities for ðk, sÞ-Riemann-Liouville fractional integral operators were obtained by Agarwal et al. [28]. The authors in [29] established certain Polya-Szego type inequalities involving generalized Katugampola fractional integral operator.
Motivated by the above works and discussions, in this manuscript, we establish certain novel inequalities for convex functions under a new generalized fractional integral operator which integrates the two proposed fractional integral operators in [30,31]. Moreover, we consider certain special cases of our main results. The results obtained extend and generalize some of the existing results by substituting some parameters.
The manuscript is marshaled as follows: Section 2 presents some main definitions and results. The acquired results are presented in Section 3. The last section concludes the manuscript.

Preliminaries
Let us first present the essential definitions and properties of fractional analysis that will be frequently used in this study.
Theorem 4 (see [32]). Let u and v be two positive continuous functions with u ≤ v on J. Assume that the functions u/v and 2 Journal of Function Spaces u are decreasing and increasing, respectively. If the function Ω is a convex with Ωð0Þ = 0, then Theorem 5 (see [32]). Let u, z, and v be positive continuous functions with u ≤ v on J: Let also u, z are increasing functions and u/v is a decreasing function. If the function Ω is a convex with Ωð0Þ = 0, then

Main Results
In this section, we will establish some new tolerances for the convex functions under a new fractional integral operator which combine together the two operators proposed in [30,31].
Proof. Thanks to the hypotheses of theorem, Ω is convex with Ωð0Þ = 0. Thus, ΩðxÞ/x is an increasing function. Furthermore, since u is increasing function, the function ΩuðxÞ/uðx Þ is increasing. Distinctly, uðxÞ/vðxÞ is decreasing function.

Conclusions
In this work, we have established certain Pólya-Szegö inequalities by using convex functions under a new generalized fractional integral operator. More precisely, some new results have been established by merging the ðk, sÞ-Riemann-Liouville fractional integral operator with the generalized Katugampola fractional integral operator. Moreover, we have introduced several new special results that cover many classical fractional integral operators. In future work, it will be very interesting to study the inequalities considered in this work under a more general fractional integral operator in terms of another function ψ, precisely, we hint to ψðxÞ = x s+1 , and this is what we will think about in the next work.

Data Availability
Data are available upon request.

Conflicts of Interest
No conflicts of interest are related to this work.