On Fractional Ostrowski-Mercer-Type Inequalities and Applications

: The objective of this research is to study in detail the fractional variants of Ostrowski– Mercer-type inequalities, speciﬁcally for the ﬁrst and second order differentiable s -convex mappings of the second sense. To obtain the main outcomes of the paper, we leverage the use of conformable fractional integral operators. We also check the numerical validations of the main results. Our ﬁndings are also validated through visual representations. Furthermore, we provide a detailed discussion on applications of the obtained results related to special means, q -digamma mappings, and modiﬁed Bessel mappings.


Introduction
In 1937, Alexandar Markowich Ostrowski [1] discovered an integral inequality known as the Ostrowski inequality, stated as: Let ψ : I = [µ, κ] ⊆ R → R (Real numbers) be a differentiable mapping on the interior of I such that ψ is integrable on [µ, κ], where µ, κ ∈ I with µ < κ.If |ψ (λ)| ≤ M for all λ ∈ (µ, κ) and M > 0, then holds for all ∈ [µ, κ], and 1  4 is the best possible constant.Ostrowski's inequality provides an approximation of the difference between mapping values and their integral average over a given interval.For more than 5000 years, inequalities have been seen in wide applications.The oldest was recorded in ancient Chinese mathematics, called the He Chengtian inequality [2].By utilizing this inequality, He Chengtian calculated the approximate values of the fractional day of a moon and a year.Over the course of time, researchers have broadened the scope of convex mappings, leading to the discovery of various variants of the Hermite-Hadamard inequality, see [3][4][5][6][7][8][9][10][11][12][13][14].
The class of convex mappings is regarded as cornerstone of the theory of inequalities with a wide range of applications in many areas of mathematics such as in numerical integration, special means and special functions.
For s-convex mapping, ψ : [µ, κ] ⊆ R → R, φ 1 , φ 2 ∈ [µ, κ] and s ∈ (0, 1], the Hermite-Hadamard-Jensen-Mercer inequality is given in [24] as follows: Fractional calculus, dealing with integrals and derivatives of arbitrary real order, has significantly contributed to the characterization of diverse real materials, such as polymers.The fractional models are more adequate than the previous used models of integer orders, see [25][26][27].In addition to classical derivatives, fractional order derivatives offer superior capabilities in describing the memory and hereditary characteristics of diverse processes.In [28], Podlubny discussed various applications of fractional derivatives.Riemann, Liouville, Grünwald and other researchers defined the fractional derivatives in several ways given in [28,29].
To investigate the characteristics of fractional differentiability and local scaling, the fractional derivatives were not suitable because of their non-local nature [30].By renormalization of the Riemann-Liouville definition, Kolwankar and Gangal [30,31] proposed the idea of local fractional derivatives.The calculus of fractal space-time is studied with the help of local fractional derivatives.In addition, two-scale fractal theory is utilized to study problems involving porous media and unsmooth boundaries [32][33][34].Moreover, the fractional derivatives are also utilized to find the approximate solutions of the fractional differential equations, see [35,36].
Probably the most frequently used definition of fractional integrals is due to B. Riemann and J. Liouville, commonly known as the Riemann-Liouville fractional integrals, defined as follows: Let ψ be an integrable mapping on [µ, κ].Then, the left and right sided Reimann-Liouville fractional integrals I ω µ + ψ and I ω κ − ψ of order ν > 0 with µ ≥ 0 are defined by: and where Γ(ν) is the gamma mapping defined as: For more details, see [37].
When the fractional 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.In the literature, there are various fractional operators with local, nonlocal, singular and non-singular kernels, [38][39][40][41].Jarad et al. [42] defined conformable fractional integrals and derivatives with two parameters and kernels, which are helpful to the better understanding of the complexity of fractional variational problems, optimal control problems and modelling of complex systems.
Let us recall beta mapping or Euler integral of the first kind with two variables defined by: In terms of gamma mapping, it is defined as: . Some properties of beta function are: 1.The beta function is symmetric i.e., B(µ 1 , κ 1 ) = B(κ 1 , µ 1 ).

2.
B(µ 1 + 1, κ 1 ) = B(µ 1 , κ 1 ) The motivation of this paper is to establish several new fractional variants of Ostrowski-Mercer-type inequalities using the first and the second order s-convex mappings of second sense.To achieve this goal, we employ conformable fractional integral operators.The main results' relevance has also been analyzed numerically and graphically.In addition, we also demonstrate some applications to means, q-digamma mappings, and modified Bessel mappings.

Ostrowski-Mercer-Type Inequalities for the First Order Differentiable s-Convex Mappings
In this section, we first establish a key lemma for the first differentiable mappings involving conformable fractional integrals.Then, by utilizing this result, we obtain several inequalities for the first order differentiable mappings whose absolute values are s-convex in the second sense.
Proof.Using Lemma 1 and the Jensen-Mercer inequality with the s-convexity of |ψ | on [µ, κ], we obtain The proof is completed.
Proof.Using Lemma 1 and the Hölder inequality for integrals, we have Now, by applying the Jensen-Mercer inequality with the s-convexity of |ψ | q , we have The proof is completed.
Proof.Using Lemma 1, power mean inequality and the Jensen-Mercer inequality with the s-convexity of |ψ | q , we have The proof is completed.
Theorem 4. For a differentiable mapping ψ : [µ, κ] → R on (µ, κ) and if |ψ | q is an s-convex mapping in the second sense on [µ, κ] with p, q > 1 and 1 p + 1 q = 1.Then, under the assumptions of Lemma 1, the following inequality holds: Proof.Taking modulus of Lemma 1 and using Young's inequality, i.e., xy ≤ 1 p x p + 1 q y q (equality holds when x p = y q ), we have Now, applying the Jensen-Mercer inequality with the s-convexity of |ψ | q , we obtain The proof is completed.
Theorem 5.For a twice differentiable mapping ψ : [µ, κ] → R on (µ, κ) and if |ψ | is an s-convex mapping in the second sense on [µ, κ].Then, under the assumptions of Lemma 2, the following inequality holds: where Proof.Using Lemma 2 and the Jensen-Mercer inequality with the s-convexity of |ψ |, we obtain The proof is completed.
Corollary 16.If we set ω = 1 and ν = 1 in Theorem 5, we obtain Theorem 6.For a twice differentiable mapping ψ : [µ, κ] → R on (µ, κ) and if |ψ | q 1 is an s-convex mapping in the second sense on [µ, κ] and p 1 , q 1 > 1.Then, under the assumptions of Lemma 2, the following inequality holds: where Proof.Using Lemma 2 and the Hölder inequality for integrals, we have . Now, by applying the Jensen-Mercer inequality with the s-convexity of |ψ | q 1 , we have The proof is completed.
Proof.Using Lemma 2, power mean inequality and the Jensen-Mercer inequality with the s-convexity of |ψ | q 1 , we have The proof is completed.
Next in Figure 1, we present the graphical visualization of Theorems 1 and 5.For this we consider the above mentioned assumptions and s ∈ (0, 1] and ω ∈ (0 Now from Theorem 2, we have 0.04834 < 0.27498 and from Theorem 6, we have 0.04377 < 0.13139.This proves the numerical validation of these results. In Figure 2, we present the graphical visualization of Theorems 2 and 6.For this we consider the above mentioned assumptions and s ∈ (0, 1] and ω ∈ (0  (30), respectively.Clearly one can see that the inequalities (15) and (30) hold good by varying both the parameters s and ω.
Now from Theorem 3, we have 0.04834 < 0.12172 and from Theorem 7, we have 0.04377 < 0.09022.This proves the numerical validation of these results.
Next in Figure 3, we present the graphical visualization of Theorems 3 and 7.For this we consider the above mentioned assumptions and s ∈ (0, 1] and ω ∈ (0  (19) and (b) (32), respectively.Clearly one can see that the inequalities (19) and (32) hold good by varying both the parameters s and ω.
Now from Theorem 4, we have 0.04834 < 0.78191 and from Theorem 8, we have 0.04377 < 0.28330.This proves the numerical validation of these results.Now in Figure 4, we present the graphical visualization of Theorems 4 and 8.For this we consider the above mentioned assumptions and s ∈ (0, 1] and ω ∈ (0   (34), respectively.Clearly one can see that the inequalities (21) and (34) hold good by varying both the parameters s and ω.

Applications
In this section, we will discuss some applications of our results.
The arithmetic mean

Modified Bessel Function
In [46], the modified Bessel mapping of the first kind ξ ω (γ) is given as follows: where γ ∈ R and ω < −1.

Conclusions
To summarize, this research study introduces new fractional versions of Ostrowski-Mercer-type inequalities by using the first and the second order differentiable s-convex

Figure 1 .
Figure 1.In the above figures, the yellow and purple surfaces show the right and left sides of inequalities (a)(13) and (b) (28), respectively.Clearly one can see that the inequalities (13) and (28) hold good by varying both the parameters s and ω.

Figure 2 .
Figure 2. In the above figures, the green and purple surfaces show the right and left sides of inequalities (a)(15) and (b)(30), respectively.Clearly one can see that the inequalities(15) and(30)

Figure 3 .
Figure 3.In the above figures, the pink and purple surfaces show the right and left sides of inequalities (a)(19) and (b)(32), respectively.Clearly one can see that the inequalities(19) and(32) hold good