New Inequalities Using Multiple Erd é lyi–Kober Fractional Integral Operators

: The role of fractional integral inequalities is vital in fractional calculus to develop new models and techniques in the most trending sciences. Taking motivation from this fact, we use multiple Erd é lyi–Kober (M-E-K) fractional integral operators to establish Minkowski fractional inequalities. Several other new and novel fractional integral inequalities are also established. Compared to the existing results, these fractional integral inequalities are more general and substantial enough to create new and novel results. M-E-K fractional integral operators have been previously applied for other purposes but have never been applied to the subject of this paper. These operators generalize a popular class of fractional integrals; therefore, this approach will open an avenue for new research. The smart properties of these operators urge us to investigate more results using them.


Introduction and Motivation
Integral inequalities are a crucial tool in basic mathematical analysis.Various names of fundamental inequalities can be found in the literature, for example, Cauchy-Schwarz, H .. older, and Minkowski inequalities.Moreover, in the past few years, fractional integral inequalities have emerged as one of the most practical and extensive instruments for the advancement of numerous topics in both pure and applied mathematics [1].Therefore, several scholars have presented a variety of generalized inequalities involving fractional integral operators [2][3][4][5][6][7][8][9][10][11].In addition to its generalizations and extensions in one or more variables, fractional calculus covers a variety of important problems involving unique mathematical physics functions and offers various potentially helpful methods for solving differential and integral equations [12].Several extensions of the Riemann-Liouville (R-L) and Erdélyi-Kober (E-K) operators have been studied in the literature.These extensions include the Bessel function J µ , as well as the Hand G-functions in the integrand [13].The work of Srivastava is mentionable for developing a rigorous and more general theory of the operators of fractional calculus.For example, many generic families of operators of fractional integration, including Fox's H-function and its extensions in two or more variables, are discussed in [14] (see also [15]).A distinct class of fractional calculus operators and their uses concerning higher transcendental functions have been examined in [16].Furthermore, numerous variations in both parameters and arguments for the fractional calculus operators, together with associated special functions and integral transformations, are offered in [17].Among these, we apply the multiple Erdélyi-Kober fractional operators [12][13][14] in this research to explore the new fractional inequalities that generalize the earlier works cited in [11].To the best of our knowledge, these multiple operators have never been used for this purpose.Therefore, before continuing with further discussion about our new results, we first present the basic preliminaries and required facts in the subsequent section.
The plan of this paper is as follows: After presenting the necessary definitions and required facts in Section 2, we proceed to prove the reverse Minkowski inequalities using multiple Erdélyi-Kober fractional operators in Section 3. A novel and new class of inequalities using the multiple Erdélyi-Kober fractional integral operator is presented in Section 4. Finally, Section 5 contains a summary of the results.

Multiple Erdélyi-Kober Fractional Integral Operators
This section is complimentary to this research as it contains all necessary preliminaries, concepts, and definitions about M-E-K integral operators.The Fox-H-function is defined by (see [12][13][14]).
The poles of the gamma function in the numerator of the above equation are split by making use of an appropriate contour L. Furthermore, the H-function reduces to the Meijer G-function [12] by considering the value of all A i = 1 = B j .The Meijer G-function is further related to many other special functions like Fox-Wright, hypergeometric, and Mittag Leffler functions, which makes the operators defined in the following definition very significant.Definition 1.Multiple Erdélyi-Kober fractional integral operators, I In the above equation, the various parameters used, i.e., (δ i ≥ 0), provide the order of integration, while (γ i ≥ 0) are multi-weight, whereas (β i ≥ 0) are also some additional multi-parameters.We can also note that ∑ m k=1 δ k > 0 and for all δ i = 0, we obtain the identity operator, i.e., I (γ k ),(0,...,0) (β k ),m f (x) := f (x), from the above equation.
Under such circumstances, the generalized fractional integrals can be broken down into commutative products of conventional operators (namely, operators).Consequently, the generalized fractional calculus with a well-developed thorough theory and numerous established applications combines the capabilities of the special functions with the widespread use of conventional fractional calculus [12,13].Similar to this, a specific number of compositions of R-L and E-K operators are also taken into consideration [18,19].H m,0 m,m (or H m,m n+n,m+n if compositions of left as well as right operators are taken) serve as the kernels for these compositions.In [12], several subjects including classes of differential and integral equations, geometric function theory, special functions, integral transformations, and operational calculus have all been addressed using the proposed theory.
In [20], Dimovski presented the spaces C µ ([0, ∞]) over real variable x > 0 of good functions.This is where we work in this study.
More properties of such spaces can be seen in [12].Similarly, the elements of Lebesgue integrable spaces (L < ∞ also preserve the power function.Furthermore, our results depend on the following assumptions: (3) Remark 1.It is evident that the kernel in the above definition of the multiple E-K fractional integral remains positive [21][22][23].
Lemma 1.For z µ = f ∈ C µ , we have the following useful result: If we consider µ = 1 in Lemma 1, then we obtain the following relation: , Furthermore, the class of weighted analytic functions is also preserved using these operators [20].These operators are bilinear, commutative, invertible, and satisfy a semigroup property [20].It is also proved that under the assumptions (3), they act as bounded linear operators over L p µ .For f ∈ C (N) µ under the assumptions in (3), M-E-K satisfy the initial conditions given by The Caputo and Riemann-Liouville versions of the M-E-K fractional operators are also investigated in [20].
Another interesting feature of these operators is how they relate to a wide range of widely used fractional integrals, as listed below [12,13], when all β ′ k s are equal, i.e., The Meijer G-function of the form can be used to describe the multiple E-K fractional integrals more simply (see [12], Chapter 1).

4.
For m = 1 = β, it reduces to the E-K fractional operator.
Furthermore, these operators also contain other well-known fractional integrals, such as Weyl, and the Hadamard and Katugampola can also be obtained (see [28] and references therein).
Section 2 specifies that the conditions on the parameters will be considered typical unless specifically stated otherwise during this investigation.

Reverse Minkowski Inequalities Using Multiple Erdélyi-Kober Fractional Operator
The M-E-K fractional operators are used in this section to state and prove reverse Minkowski integral inequalities, and the following theorem is our main result about the reverse Minkowski fractional integral inequalities.
Proof.For t ∈ [0, x]; x > 0, and Ξ(t) Θ(t) < M, we obtain the following inequality: Next, for x > 0 we take the following expression: Then, by multiplying it on both sides of (8) because of Remark 1 and integrating the resultant inequality for t ∈ [0, x]; x > 0, we obtain This implies that and Next, by making use of the inequality mΘ(t) ≤ Ξ(t), one can obtain leading to the following: Now, if we multiply (9) with the above expression (11) and integrate the resultant inequality for t ∈ [0, x]; x > 0, we obtain We can compute the required result as stated in Theorem 1 with the addition of inequality (10) and inequality (12).
. Furthermore, for x > 0, consider two positive functions Ξ and Θ on the interval [0, ∞) satisfying, I Proof.We can achieve this in two steps.Firstly, we will multiply the inequalities ( 10) and ( 12).This will lead to the following: In the second step, we achieve the subsequent result by making use of the well-known Minkowski inequality on the right-hand side (RHS) of ( 14): Hence, the required result, i.e., (13), of the stated theorem follows from inequalities ( 14) and (15).□

New Inequalities Using Multiple Erdélyi-Kober Fractional Integral Operator
This section contains the proof of the novel inequalities involving the M-E-K operators.
Theorem 3.For 1 r + 1 s = 1; r, s > 1, consider Ξ, Θ to be a pair of positive functions on the interval [0, ∞) such that I Proof.For t ∈ [0, x], x > 0 and Ξ(t) Θ(t) ≤ M < ∞, we obtain the following: It will lead to the following: Next, we multiply ( 9) and ( 18) then integrate the result for t ∈ [0, x], x > 0 This implies that Finally, we compute Furthermore, by considering mΘ(t) ≤ Ξ(t), we obtain This implies that Next, we multiply (23) and ( 9) and then integrate the result for t ∈ [0, x], x > 0 to obtain Therefore, we have The required result is obtained when we multiply Equations ( 21) and (25).□ Theorem 4. Consider 1 r + 1 s = 1; r, s > 1 and a pair of positive functions Ξ, Θ on the interval [0, ∞) satisfying I then the following inequality can be obtained: Proof.For x > 0; t ∈ [0, x], by making a replacement of Ξ(t) and Θ(t) with Ξ(t) r and Θ(t) s , in Theorem 3, we can compute the required result (26).□ Theorem 5. Consider a pair of positive functions Ξ and Θ on [0, ∞), where Ξ is non-decreasing and Θ is non-increasing.Therefore, we can obtain where I is defined by (5) and λ, ν > 0.
Proof.Suppose λ > 0, ν > 0, and t, u ∈ [0, x], x > 0; then, we obtain This implies that We can integrate the product of ( 29) and ( 9) for the variable t on the interval [0, x] to obtain the following This implies that A product of ( 9) and (31), after integration over the variable u ∈ [0, x], yields which leads to the required result.□ Theorem 6.Consider a pair of positive functions Ξ and Θ on the interval [0, ∞) so that Ξ is non-decreasing and Θ is non-increasing.These assumptions will lead to the following statement: where x > 0; λ; ν > 0 and I Proof.Consider the product of with (31) and then integrate the result for the variable u ∈ (0, x) to obtain the following: This provides the required result (33).□ Remark 2. The stated inequalities ( 27) and (33) can be reversed by making use of the functions Remark 3. Considering Theorem 6 using ε k = γ k and η k = δ k , we obtain the statement of Theorem 5.
Theorem 7. Consider a pair of functions such that Ξ ≥ 0 and Θ ≥ 0 on the interval [0, ∞) so that Θ is non-decreasing.Then, the assumption leads to the following result: Proof.Since, λ > 0, ν > 0; therefore, by making use of an arithmetic-geometric inequality, we obtain Next, by considering the product of ( 9) and (38) and integrating it for the variable t ∈ (0, x), we obtain This will lead to the following: which can be expressed as This implies that Next, by making use of (36), we obtain which leads to the desired result.□ Theorem 8. Consider three positive functions, namely, Ξ, Θ, and h, to also be continuous on the Then, for every x > 0, we state that Proof.Given that Ξ, Θ, and h are positive as well as continuous functions on the interval [0, ∞), by making use of (41), we obtain Consider the product of (43) with h(t)h(u); then, we have Next, an integration of the product of ( 9) and (44) for the variable t ∈ (0, x) leads to the following: This implies that A product of (45) with F(x, u) leads to the subsequent expression after integration over u Therefore, we obtain This leads to the required result.□ Theorem 9. Consider three positive functions Ξ, Θ, and h to be continuous over [0, ∞) so that Then, we can state that Proof.Consider the product of (31) with (34) and obtain the subsequent result after integrating this product over u ∈ (0, x): This implies that which completes the steps of the required proof.□ Remark 4.An application of the Theorem 9 by replacing ε k with γ k and η k with δ k leads to the statement of Theorem 8.
Theorem 10.Consider a pair of positive continuous functions, namely Ξ and h, satisfying Ξ ≤ h over the interval [0, ∞).Suppose Ξ h is decreasing while Ξ is increasing over the interval [0, ∞).Then, for all x > 0 as well as for any λ ≥ 1, we state that Proof.A substitution of Θ = Ξ λ−1 in the statement of Theorem 8 leads to the following: The inequality Ξ ≤ h leads to the following: Integrating the product of ( 9) and (51) over the interval t ∈ (0, x), we obtain which leads to the following: Making use of (52), we state that 1 , and therefore we obtain Hence, a combination of (50) and (53) leads to the required result.□ Theorem 11.Consider a pair of positive continuous functions, namely Ξ and h, satisfying Ξ ≤ h over the interval [0, ∞).Suppose Ξ h is decreasing and Ξ is increasing on [0, ∞), then for all x > 0 as well as for any λ ≥ 1, we obtain Proof.Substituting Θ = Ξ λ−1 in the statement of Theorem (9), we obtain According to our assumption Ξ ≤ h, we obtain the following: Next, considering the product of F(u, t) as defined in ( 9) with ( 56) and then integrating it over u ∈ (0, x) will lead to the following result: Next, considering the product of (57) with I (60) Finally, (55) and (60) lead to the required result.□ Theorem 12.For all t, u ∈ (0, x), x > 0, we consider three positive functions Ξ, Θ, and h to also be continuous over the interval [0, ∞) satisfying (Ξ(t) − Ξ(u))(Θ(t) − Θ(u))(h(u) + h(t)).
In this way, we obtain the following: Proof.Following the conditions of Theorem