Some Double Generalized Weighted Fractional Integral Inequalities Associated with Monotone Chebyshev Functionals

In this manuscript, we study the unified integrals recently defined by Rahman et al. and present some new double generalized weighted type fractional integral inequalities associated with increasing, positive, monotone and measurable function F . Also, we establish some new doubleweighted inequalities, which are particular cases of the main result and are represented by corollaries. These inequalities are further refinement of all other inequalities associated with increasing, positive, monotone and measurable function existing in literature. The existing inequalities associated with increasing, positive, monotone and measurable function are also restored by applying specific conditions as given in Remarks. Many other types of fractional integral inequalities can be obtained by applying certain conditions on F and Ψ given in the literature.


Introduction
In the context of fractional differential equations, integral inequalities are very significant. This field has gained popularity during the last few decades. Various researchers, such as [1][2][3], have investigated the significant developments in this domain. By employing Riemann-Liouville (R-L) fractional integrals, the authors presented Grüss type and several other new inequalities in [4,5]. Certain inequalities for the generalised (k, ρ)-fractional integral operator are proposed in [6]. In [7], the modified Hermite-Hadamard type inequalities can be found. Dahmani [8] discovered various fractional integral inequalities employing a family of n positive functions. In [9], Srivastava et al. presented the Chebyshev inequality by employing general family of fractional integral operators. Some remarkable inequalities and their applications can be found in nthe work of [10][11][12][13][14][15].
In [16,17], the Chebyshev functional for the integrable functions Z 1 and Z 1 on [v 1 , v 2 ], is given by (1) where the functionh 1 is a positive and integrable on [v 1 , v 2 ].
The following extended Chebyshev functional for the integrable functions Z 1 and Z 1 on [v 1 , v 2 ] can be found in [5,18] by where the two functionsh 1 andh 2 are positive and integrable on [v 1 , v 2 ].

Some Double-Weighted Generalized Fractional Integral Inequalities
In this section, some double-weighted generalized fractional integral inequalities are presented. To this end, we begin by proving the following Lemma. Lemma 1. Let the function F be measurable, increasing, positive and monotone function on (v 1 , v 2 ), and has a continuous derivative Proof. Assume that Z 1 : . Then, one may gets

Consequently, it follows
By utilizing the given condition v 1 ≤ ≤ θ ≤ ξ ≤ v 2 , we get Applying (11) for the particular case when Z 1 (v) = v, then we can write Thus with the aid of (7), the above equation gives, which completes the proof.
Based on Lemma 1, we prove the following theorem.

Theorem 4.
Suppose that the function F be measurable, increasing, positive and monotone function on (v 1 , v 2 ), and has a continuous derivative Proof. By employing the definition (7) and Lemma 1, we obtain Consequently, it follows that By applying Cauchy-Schwartz inequality [40], we get Hence, using (11) and (13) concludes the proof.
The following new particular results of Theorem 4 can be easily obtained.

Corollary 1.
Suppose that the function F be measurable, increasing, positive and monotone function on (v 1 , v 2 ), and has a continuous derivative . Then, we have in Theorem 4, the desired result is obtained.

Corollary 2.
Suppose that the function F be measurable, increasing, positive and monotone function on (v 1 , v 2 ), and has a continuous derivative in Theorem 4, desired corollary is proven.

Corollary 3.
Suppose that the function F be measurable, increasing, positive and monotone function on (v 1 , v 2 ), and has a continuous derivative . Then, we have , then Theorem 4 and Corollaries 1-3 will reduce to the work of Bezziou et al. [27].
Theorem 5. Let F be measurable, increasing, positive and monotone function on (v 1 , v 2 ), and having a continuous derivative Proof. Consider the left-hand side of (14), we have Applying Cauchy-Schwartz inequality [40] to the above equation yields, In view of (7), we get the desired proof of (14).
. Then, we have , the desired result is obtained.
Theorem 6. Let F be measurable, increasing, positive and monotone function on (v 1 , v 2 ), and having a continuous derivative F on (v 1 , v 2 ). Assume that f 1 : Proof. Consider the left-hand side of (15), we have Hence taking (7) into account, the proof of (15) is completed.
Theorem 7. Let F be measurable, increasing, positive and monotone function on (v 1 , v 2 ) and having continuous derivative Proof. Consider the left-hand side of (16), we have Hence, by using (7), the proof of the theorem is completed.

Concluding Remarks
In the study of mathematics and related subjects, mathematical inequalities are extremely important. Fractional integral inequalities are now useful in determining the uniqueness of fractional partial differential equation solutions. They also guarantee the boundedness of fractional boundary value problem solutions. These suggestions have promoted the future research in the subject of integral inequalities to investigate the extensions of integral inequalities using fractional calculus operators. In the present investigation, we have proposed some double-weighted generalized fractional integral inequalities by utilizing more generalized class of fractional integrals associated with integrable, measurable, positive and monotone function F in its kernel. The derived inequalities are more general than the existing inequalities cited therein. All the classical inequalities can be easily restored by applying specific conditions on F and Ψ(θ) given in Remark 3. Also, we can derive some new weighted type double fractional integral inequalities by applying specific conditions on F and Ψ(θ) given in Remark 2. In future research, some new other type of inequalities will be derived by employing the proposed operator. The special cases of the obtained result can be found in [24,25,27,41,42].