Certain New Chebyshev and Grüss-Type Inequalities for Unified Fractional Integral Operators via an Extended Generalized Mittag-Leffler Function

In this paper, by adopting the classical method of proofs, we establish certain new Chebyshev and Grüss-type inequalities for unified fractional integral operators via an extended generalized Mittag-Leffler function. The main results are more general and include a large number of available classical fractional integral inequalities in the literature. Furthermore, some new fractional integral inequalities similar to the main results can be also obtained by employing the newly introduced generalized fractional integral operators involving the Mittag-Leffler-like function and weighted function. Consequently, their relevance with known inequalities for different kinds of fractional integral operators are pointed out.


Introduction
Let f and g be two continuous and synchronous functions on [a, b], that is, the two continuous functions f and g satisfying ( f (x) − f (y))(g(x) − g(y)) ≥ 0 for x, y ∈ [a, b]. Then the following inequality holds (1) The reverse inequality holds always whenever f and g are two continuous and asynchronous functions. The foregoing inequality (1) is called as the well-known Chebyshev integral inequality. Over the last several years, by employing various kinds of fractional integral operators, many researchers have extended the classical inequalities to fractional integral inequalities at home and abroad, we refer the reader to [1][2][3][4][5][6] and the references quoted therein. For example, using the Saigo fractional integral operators, Khan et al. [7] presented some inequalities for a class of n-decreasing positive functions. With the help of fixedpoint theorems and inequalities analysis techniques, Baleanu et al. [8] and Khan et al. [9] investigated the existence results for hybrid fractional differential equation boundary value problems, respectively. At present, there have been a great deal of fractional integral operators and their applications introduced in the books [10,11]. Furthermore, Belarbi and Dahmani [12] used the Riemann-Liouville fractional integrals to present the Chebyshevtype integral inequalities. In other words, if f and g are two synchronous functions on C[0, +∞), then, for x > 0 and α > 0, the following two inequalities hold and for x > 0 and α, β > 0, where R α and R β denote the Riemann-Liouville fractional integrals of order α and β, respectively. Similar to the inequalities (2) and (3), Ögünmez and Özkan [13], Chinchane and Pachpatte [14], Purohit and Raina [15], Habib et al. [16] and Set et al. [17] investigated the Chebyshev-type inequalities for the Riemann-Liouville fractional q-integral operators, the Hadamard fractional integral operators, the Saigo fractional integral and q-integral operators, and generalized k-fractional conformable integrals, respectively. Here it is easy to see that the Riemann-Liouville fractional integral and q-integral operators can be seen as the special case of the Saigo fractional integral and q-integral operators, respectively. By applying the Riemann-Liouville fractional integral operators, Dahmani [18] obtained the following weighted fractional Chebyshev-type integral inequalities, which are the extensions of inequalities (2) and (3). Under the same conditions of inequalities (2) and (3), furthermore, let u, v : [0, ∞) → [0, ∞) be continuous. Then we have (4) and (5) for x > 0 and α, β > 0. Similar to inequalities (4) and (5), Chinchane and Pachpatte [19,20], Brahim and Taf [21], Yang [22,23] and Liu et al. [24] studied the weighted fractional Chebyshev-type integral inequalities for Hadamard and Saigo fractional integral operators, fractional integral operators with two parameters of deformation q 1 and q 2 , fractional q-integral operators, Saigo fractional integral and q-integral operators, and generalized fractional integral operators involving the Gaüss hypergeometric function. respectively. In the book [25], the following inequality is provided: (6) where f and g are two integrable functions on [a, b] satisfying the following conditions Here inequality (6) is well-known Grüss inequality. It has attracted extensive attention of scholars all over the world. For example, Elezović et al. [26] derived some Grüss type inequalities related to Chebyshev functional under the function spaces L p with weight function and exponents. Liu and Ngô [27] gave the inequality of Ostrowski-Grüss type on time scales, which unified corresponding continuous, discrete and quantum calculus versions. Dragomir [28] established some sharp Grüss type inequalities for functions with bounded variation and selfadjoint operators in Hilbert space. Furthermore, Dragomir [29] obtained some Grüss type inequalities for the complex integral under various assumptions.
When f , g satisfy the conditions (7), Dragomir [30] proved the following inequality where u and v are two nonnegative continuous functions on [a, b] and the Chebyshev functionals were defined as and When f , g ∈ L ∞ (a, b), Dragomir [30] had the following inequality Furthermore, let f be M-g-Lipschitzian on [a, b], i.e., then the following inequality holds [30] |S Let f and g be L 1 and L 2 -lipschitzian functions on [a, b], respectively; Dragomir [30] provided the inequality Similar to inequality (6), Dahmani et al. [31] and Zhu et al. [32] studied the Grüss type inequality for Riemann-Liouville fractional integral and q-integral operators satisfying the conditions (7), respectively. Similar to inequality (8), Dahmani and Benzidane [33] gave the Riemann-Liouville fractional q-integral inequality satisfying the conditions (7). Dahmani [34] obtained the fractional integral inequalities (11), (13) and (14) for the extended Chebyshev functional (10) based on the Riemann-Liouville fractional integrals. Based on the Riemann-Liouville fractional q-integral and integral operators, Brahim and Taf [21,35] established the fractional q-integral and integral inequalities (11), (13) and (14) for the extended Chebyshev functional (10) with two parameters of deformation q 1 and q 2 , respectively. By using the Saigo fractional integral and q-integral operators, the author obtained the Saigo fractional integral and q-integral inequalities (8), (11), (13) and (14) for the extended Chebyshev functional (10), respectively. Akdemir et al. [36] gave the general variants of Chebyshev type inequalities using the generalized fractional integral operators.
In 2020, Yang et al. [37] obtained the unified fractional generalized Hadamard and Fejér-Hadamard inequalities for m-convex functions containing extended generalized Mittag-Leffler function. In 2021, Zhang et al. [38] investigated some inequalities for unified fractional integral operators via strongly (α, h − m)-convex function. In 2021, Jung et al. [39] studied the refinements of some integral inequalities for unified fractional integral operators. Motivated by the works mentioned earlier, the main aim of this paper is to establish certain new Chebyshev and Grüss-type inequalities for unified fractional integral operators via an extended generalized Mittag-Leffler function by using the classical method of proofs In Section 2. In Section 3, we show that the unified fractional integral operators contains a lot of existing fractional integral operators. We also introduce two newly generalized fractional integral operators involving the Mittag-Leffler-like function and weighted function. Using the newly introduced generalized fractional integral operators, some new fractional integral inequalities can be also obtained. Furthermore, their relevance with known inequalities for different kinds of fractional integral operators are pointed out. The main results of this paper are more general and include a great number of available classical inequalities in the literature.
Definition 1 (See [40,41] where R(µ) denotes the real part of complex number, Γ and B represent the Gamma and Beta functions, respectively. Here the generalized Pochhammer symbol (c) nk and an extension of the beta function B p are defined as follows: where R(x), R(y), R(p) > 0.
In this section, we nextly give some new Chebyshev-type integral inequalities for the synchronous functions involving the left unified fractional integral operators.  (20) Proof. Since f and g are two synchronous functions on [a, b], then for all τ > 0 and ρ > 0, we have It follows from (21) that we write Multiplying both sides of (22) by v(τ)ξ (τ)K τ x (E γ,δ,k,c µ,α,l , ξ; φ) and integrating the obtained inequality with regard to τ from a to x, we get Multiplying both sides of (23) by u(ρ)ξ (ρ)K ρ x (E γ,δ,k,c µ,α,l , ξ; φ) and integrating the obtained inequality with regard to ρ from a to x, we obtain (24) which implies (20).
Proof. Putting u = v, v = w and using Lemma 1, we can write Multiplying both sides of (26) by F φ u (x), we obtain Putting u = u, v = w and using Lemma 1, we can write Multiplying both sides of (28) by F φ v (x), we obtain With the same arguments as before, we can get The required inequality (25) follows on adding the inequalities (27), (29) and (30).
Proof. Putting u = v, v = w and using Lemma 2, we can write Multiplying both sides of (35) by F φ u (x), we obtain Putting u = u, v = w and using Lemma 2, we can write Multiplying both sides of (37) by F φ v (x), we obtain With the same arguments as before, we can get The required inequality (34) follows on adding the inequalities (36), (38) and (39).
Proof. Since f and g are two synchronous functions on [a, b] and let h and u be two nonnegative continuous functions on [a, b], then for all τ > 0 and ρ > 0, we have Expanding the left hand side of (41) that we write . (42) Multiplying both sides of (42) by u(τ)ξ (τ)K τ x (E γ,δ,k,c µ,α,l , ξ; φ) and integrating the obtained inequality with regard to τ from a to x, we get Multiplying both sides of (43) by u(ρ)ζ (ρ)K ρ x (Eγ ,δ,k,ĉ ν,β,ι , ζ; ϕ) and integrating the obtained inequality with regard to ρ from a to x, we obtain (44) which implies (40).
and let u be a nonnegative continuous function on [a, b]. Then the following inequality holds Proof. The proof is similar to that given in Theorem 3.
Theorem 5. Let f , g be two integrable functions on [a, b] and let u be a nonnegative continuous function on [a, b]. Then the following inequality holds Proof.
Multiplying both sides of (52) by and integrating the obtained inequality with regard to τ and ρ from a to x, respectively, we get (53) which implies (49).
Theorem 6. Suppose that f , g are two integrable functions satisfying the condition (7) on [a, b] and let u, v be two nonnegative continuous functions on [a, b]. Then the following inequality holds x (Eγ ,δ,k,ĉ ν,β,ι , ζ; ϕ) and integrating the resulting result with respect to τ and ρ from a to x, respectively, we can get (58) According to the condition (7), we have Combining (58) and (59), we obtain that This ends the proof.
Theorem 7. Suppose that f , g are two integrable functions satisfying the condition (12) and let u, v be two nonnegative continuous functions on [a, b]. Then the following inequality holds Proof. From the condition (12), we have According to (57) and (62), we obtain Combining (58) and (63), we get that This completes the proof.
Theorem 8. Suppose that f , g are two integrable functions satisfying the lipschitzian condition with the constants L 1 , L 2 and let u, v be two nonnegative continuous functions on [a, b]. Then we have Proof. From the conditions of Theorem 8, we have which implies that Combining (58) and (67), we get that This ends the proof of Theorem 8.
, and the result follows from Theorem 8. This completes the proof.
and let u be a nonnegative continuous function on [a, b], p, q, r > 1 with 1/p + 1/p = 1, 1/q + 1/q = 1 and 1/r + 1/r = 1. Then the following weighted fractional integral inequality holds Proof. Multiplying both sides of (57) by and integrating the given result with respect to τ and ρ from a to x, we can state that (71) On the other hand, from (57), we have By employing the Hölder inequality, we obtain Combining (72) and (73), we get According to inequalities (71) and (74), we can write Applying the double integral Hölder inequality to (75), we obtain Using the following properties then (76) can be rewritten as which completes the desired proof.
Now we follow the proof of Theorems 6 and 9, we can get the following result.
Lemma 3. Let f be an integrable function satisfying the condition (7) and let u be a continuous function on [a, b]. Then we have the following equation Proof. Since f is an integrable function satisfying the condition Multiplying both sides of (81) by u(ρ)ξ (ρ)K ρ x (E γ,δ,k,c µ,α,l , ξ; φ) and integrating the obtained equality with regard to ρ from a to x, we have Multiplying both sides of (82) by u(τ)ξ (τ)K τ x (E γ,δ,k,c µ,α,l , ξ; φ) and integrating the obtained equality with regard to τ from a to x, we have (83) which gives (80).
Theorem 11. Suppose that f , g are two integrable functions satisfying the condition (7) and let u be a nonnegative continuous function on [a, b]. Then we have the following inequality Proof. Multiplying both sides of (57) by u(τ)ξ (τ)K τ x (E γ,δ,k,c µ,α,l , ξ; φ) and integrating the resulting identity with respect to τ and ρ from a to x, we can state Thanks to the weighted Cauchy-Schwartz integral inequality for double integrals, we can write that Thus, from (87) and Lemma 3, we get Combining (85), (86), (88) and (89), we deduce that Now using the elementary inequality 4xy ≤ (x + y) 2 , x, y ∈ R, we can state that From (90)-(92), we obtain (84). This complete the proof of Theorem 11.
Lemma 4. Let f , g be two integrable functions on [a, b] and let u, v be two nonnegative continuous functions on [a, b]. Then the following inequality holds Proof. Thanks to the weighted Cauchy-Schwartz integral inequality for double integrals, it follows from (58) that we obtain (93).

Lemma 5. Let f be an integrable function on [a, b] and let u and v be two nonnegative continuous functions on [a, b]. Then we have the following equation
Proof. Multiplying both sides of (82) by v(τ)ζ (τ)K τ x (Eγ ,δ,k,ĉ ν,β,ι,a + , ζ; ϕ) and integrating the obtained equation with respect to τ from a to y, we have which gives (95) and proves the Lemma 5.

Theorem 13. Let f be an integrable function on [a, b] and let u and v be two nonnegative continuous functions on [a, b]. Suppose that there exist two integrable functions
Then the following inequality holds: This implies that Multiplying (103) by u(τ)ξ (τ)K τ x (E γ,δ,k,c µ,α,l , ξ; φ)v(ρ)ζ (ρ)K ρ y (Eγ ,δ,k,ĉ ν,β,ι , ζ; ϕ) and integrating the given inequality with respect to τ and ρ from a to x and a to y, respectively, we can get which gives (101).
Theorem 15. Let f , g be two integrable functions on [a, b] satisfying the condition (106) and let u, v be two nonnegative continuous functions on [a, b]. Then the following inequality holds Proof. It follows from Lemma 4 that we have From Lemma 6, we can get Equations (131) and (132) together with inequality (128) yield the required equality (126).
Corollary 10. Let f and g be two integrable functions on [a, b] satisfying the condition (106) and let u be a nonnegative continuous function on [a, b]. Then the following inequality holds Furthermore, let φ = x α and u(x) = 1 in Corollary 10, we can obtain the following corollary.
Definition 4 (See [48]). Let ψ, ξ : [a, b] → R, 0 < a < b, be the functions such that ψ be positive and ψ ∈ L 1 [a, b], and ξ be differentiable and strictly increasing. Also let φ be a positive function such that φ/x is an increasing on [a, +∞). Then for x ∈ [a, b], the left and right fractional integral operators are defined as follows: Remark 10. The fractional integral operators defined as in (139) and (140) particularly produce large amounts of known fractional integral operators corresponding to different settings of the functions ξ and φ.
Similar to Definition 2, we introduce the following generalized fractional integrals.
Definition 6. Let ψ, ξ : [a, b] → R, 0 < a < b, be the functions such that ψ be positive and ψ ∈ L 1 [a, b], and ξ be differentiable and strictly increasing. Also let φ be a positive function such that φ/x is an increasing on [a, +∞) and ω ∈ R, ρ, λ, µ > 0. Then for x ∈ [a, b], the left and right generalized fractional integral operators are defined by where Ω(t) is a weighted function with Ω(t) = 0 for any t ∈ [a, b]. When σ(0) = 1, λ = α and ω = 0, the operators (143) and (144) degenerate into the left and right integral operators defined by Remark 11. The fractional integral operators defined as in (143) and (144) particularly produce several known fractional integral operators corresponding to different settings of the functions ξ and φ.

Remark 12.
Similar to the main results in Section 2, all inequalities containing the left-side generalized fractional integral operator (143) hold all the same. Furthermore, by using the proof methods of main theorems in Section 2, we can obtain all main results containing the fractional integral operators mentioned in Remarks 9-11.
Remark 13. From Remarks 9-11, we can see easily that the left and right-side unified fractional integral operators (17) and (18) as well as generalized fractional integral operators (143) and (144) involve a large number of existing fractional integral operators. Therefore, the main results of this paper can be seen as the generalizations of the existing results in the literature. For example, some specific results are given as follows.

Conclusions
In this paper, we have investigated the Chebyshev-and Grüss-type inequalities for unified fractional integral operators via an extended generalized Mittag-Leffler function. Then two generalized fractional integral operators involving the Mittag-Leffler-like function and weighted function have been introduced. Using the newly introduced generalized fractional integral operators, some new inequalities similar to the main results can be also presented. Moreover, their relevance with known inequalities for different kinds of fractional integral operators have been demonstrated. Based on main results in this paper, our future research objects are to investigate some other inequalities by using the unified fractional integral operators and generalized fractional integral operators introduced in this paper.