On Ostrowski–Mercer’s Type Fractional Inequalities for Convex Functions and Applications

: This research focuses on the Ostrowski–Mercer inequalities, which are presented as variants of Jensen’s inequality for differentiable convex functions. The main ﬁndings were effectively composed of convex functions and their properties. The results were directed by Riemann–Liouville fractional integral operators. Furthermore, using special means, q-digamma functions and modiﬁed Bessel functions, some applications of the acquired results were obtained.


Introduction
We will begin by introducing the Ostrowski inequality, which produces an upper bound for the approximation of the integral average 1 v−u v u W (k)dk by the value of W (k) at the point k ∈ [u, v] and has quite a lot of applications in the field of inequalities.
Let W : J ⊆ R → R be a differentiable mapping on J • , the interior of the interval J, such that W ∈ L 1 [u, v], where u, v ∈ J with v > u.If |W ( )| ≤ M, for all ∈ [u, v], then the following inequality holds: Many aesthetic inequalities on convex functions exist in the literature, among which Jensen's inequality has a special place.This inequality is proved under fairly simple conditions, and is extensively used by researchers in fields such as information theory and inequality theory.Jensen's inequality is presented as follows: Let 0 < χ 1 ≤ χ 2 ≤ ... ≤ χ n and σ = σ 1 , σ 2 , ..., σ n be non-negative weights such that ∑ n =1 σ = 1.The Jensen inequality (see [14]) in the literature states that if W is a convex function on the interval u, v , then holds for all χ ∈ [ u, v], σ ∈ [0, 1] and = 1, 2, ..., n.It is a crucial inequality in information theory that aids in the extraction of bounds for useful distances (see [15][16][17]).
Although many researchers have focused on Jensen's inequality, the version proposed by Mercer is the most interesting and remarkable among them.Mercer [18], in 2003, introduced a new variant of Jensen's inequality given as follows: If W is a convex function on [u, v], then holds for all χ ∈ [ u, v], σ ∈ [0, 1] and = 1, 2, ..., n.
Several refinements of Jensen-Mercer inequalities were put forth by Pečarić, J. et al. [19].Mercer's type inequalities later received many adaptations to higher dimensions by Niezgoda [20].Recently, it has made a significant addition to inequality theory, owing to its well-known characterizations.The concept of the Jensen inequality for super quadratic functions was considered by Kian [21].
The Jensen-Mercer inequality was credited to Kian and Moslehian [22], and the following Hermite-Hadamard-Mercer inequality is as follows: where W is the convex function on [u, v].
For more recent studies linked to the Jensen-Mercer inequality, one can refer to the following articles [23][24][25][26].Although fractional analysis has a history as long as classical analysis, it has recently gained popularity among researchers.It is constantly striving to advance with its use in real-world problems, contribution to engineering sciences and opportunity for development in different dimensions.One aspect that keeps fractional analysis up to date is the definition of fractional order derivatives and integrals, as well as the contribution of each new operator to different fields.When the new operators are closely examined, various features such as singularity, locality, generalization and differences in their kernel structures become apparent.Although generalizations and inferences are the foundations of mathematical methods, the new fractional operators add new features to solutions, particularly for the time memory effect.Accordingly, various operators such as Riemann-Liouville, Grünwald Letnikov, Raina, Katugampola, Prabhakar, Hilfer, Caputo-Fabirizio and Atangana-Baleanu reveal the true potential of fractional analysis.Now we will continue by introducing the Riemann-Liouville integral operators, which have a special place among these operators.
Then, Riemann-Liouville fractional integrals of order α > 0 are defined as follows: The Riemann-Liouville fractional integral operator is further expanded to many new intrgral operators, i.e., k-Riemann-Liouville fractional integral [27], ψ-Riemann-Liouville fractional integrals [28], Katugampola fractional integrals [29], k, s-Riemann-Liouville fractional integrals [30] and many such new definitions.Inspired by the Riemann-Liouville fractional integral operators, Ahmad et al. [31] introduced a new fractional integral operator involving an exponential function in its kernel and established a few generalizations of the Hermite-Hadamard type and its inequalities.It has applications in the Schrödinger Equation [32], electrical screening effect [33] and delayed nonlinear oscillator [34].
Recently, the effect of fractional analysis has begun to be felt more in the theory of inequality.Many new inequalities and new approaches for some well-known inequalities have been introduced using fractional operators.The Hermite-Hadamard inequality has been generalized with the Riemann-Liouville integral operators, which is the most important result of this effort.This generalization is presented by Sarikaya et al. as follows (see [35]).[u, v], then the following inequality for fractional integral holds: with α > 0.
The Ostrowski inequality was generalized by numerous mathematicians in various ways.In particular, a number of academic studies that consider various convexities have been published in this area.Alomari et al. [1], for instance, employed the concept of s-convexity, and Icscan et al. [41] used the concept of harmonically s-convex function.The fractional variant of the Ostrowski-type inequality was first proposed by Set [42] using Riemann-Liouville fractional operators.Liu [43] developed new iterations of Ostrowskitype inequality for the MT-convex function using the equality proved in [42].By using the Raina fractional integral operator, Agarwal et al. [44] examined a more generalised Ostrowski-type inequality.To create novel generalizations of the Ostrowski-type inequality, Sarikaya et al. [45] used local fractional integrals.For an extended form of the Ostrowski inequality, Gurbuz et al. [46] employed the Katugampola fractional operator.Atangana-Baleanu fractional operator for differentiable convex functions was used by Ahmad et al. [47] to show some innovative generalization of the Ostrowski inequality.As an advancement of this inequality, Sial et al. [48] presented Ostrowski-Mercer type inequalities for differentiable convex functions, and Ali et al. [49] used harmonically convex functions to prove new versions of Ostrowski-Mercer-type inequalities.
The major objective of this study is to create some novel Mercer-Ostrowski-type inequalities for convex functions by using Riemann-Liouville fractional integral operators with the help of a novel integral identity.Applications of the results were also presented considering numerous particular cases of the primary findings.

Main Results
In this section, we present Mercer-Ostrowski inequalities for the first differentiable functions on (u, v) for the Riemann-Liouville integral operators.For this, we introduce a new fractional identity that will act as an aid in establishing future findings.
and α > 0, the following identity holds true: Proof.Let us start with where By substituting the variables, we obtain and similarly, we get By placing the I 1 and I 2 with (9), we obtain (7).

Remark 1.
Taking then under the assumptions of Lemma 1, the following inequality holds true for all α > 0.
Proof.From Lemma 1 and the Jensen-Mercer inequality with |W | is a convex function on [u, v], we obtain which completes the proof.
Proof.Under the hypothesis of the Hölder integral inequality and the Jensen-Mercer inequality with a convexity of |W | q for Lemma 1, we obtain .
The proof is completed.
q ≥ 1, then under the assumptions of Lemma 1, the following inequality W ( ) holds true for all α > 0.
Proof.Under the assumption of the power-mean integral inequality and the Jensen-Mercer inequality with a convexity of |W | q for Lemma 1, we have W ( ) , which completes the proof.
Proof.Under the assumption of Lemma 1, we have Using Young's inequality, i.e., xy ≤ 1 p x p + 1 q y q , (equality holds if Under the assumption of Jensen-Mercer inequality and a convexity of |W | q , we obtain This concludes the proof.
Proof.Under the assumption of Hölder's inequality and Lemma 1, we have Since |W | q is a convex function, from (5), we get We obtain the following inequality ( 19) by placing inequalities (21) and ( 22) in ( 20).The proof is completed.
Proof.Under the assumptions of Corollary 5 and for W ( ) = n , we obtain the desired result.Proof.Under the assumptions of Corollary 2 and for W ( ) = n , we obtain the desired result.

Corollary 3 .
Under the assumption of |W | ≤ M, Theorem 2 gives