On generalized fractional integral inequalities for the monotone weighted Chebyshev functionals

In this paper, we establish the generalized Riemann–Liouville (RL) fractional integrals in the sense of another increasing, positive, monotone, and measurable function Ψ . We determine certain new double-weighted type fractional integral inequalities by utilizing the said integrals. We also give some of the new particular inequalities of the main result. Note that we can form various types of new inequalities of fractional integrals by employing conditions on the function Ψ given in the paper. We present some corollaries as particular cases of the main results.


Introduction
The Chebyshev functional is given by (see [7,10]) where U and V are integrable functions on [x 1 , x 2 ], and μ is a positive integrable function on [x 1 , x 2 ]. Applications of functional (1) are found in probability and statistical problems.
In the last few decades, the researchers investigated different kinds of integral inequalities by considering various integral approaches. In [14] the authors gave weighted Grüsstype inequalities by taking RL-fractional integrals into account. Dahmani [8] proposed some new inequalities in the sense of fractional integrals. Several inequalities for the extended gamma function and confluent hypergeometric k-function are found by Nisar et al. [38]. Nisar et al. [39] used Riemann-Liouville and Hadamard k-fractional derivatives and investigated Gronwall-type inequalities with applications. Rahman et al. [55] studied (k, ρ)-fractional integrals and investigated the corresponding inequalities. Sarikaya and Budak [59] proposed Ostrowski-type inequalities by considering local fractional integrals. Sarikaya et al. [60] proposed the idea of generalized (k, s)-fractional integrals with applications. Set et al. [61] investigated Grüss-type inequalities for the generalized k-fractional integrals. Recently, Jarad et al. [22,23] proposed the idea of fractional conformable and proportional fractional integral operators. Huang et al. [20] recently presented generalized Hermite-Hadamard-type inequalities for k-fractional conformable integrals. Qi et al. [45] proposed Chebyshev-type inequalities by using generalized k-fractional conformable integrals. Rahman et al. [56] investigated Chebyshev-type inequalities by utilizing fractional conformable integrals. Chebyshev-type inequalities and Minkowski-type inequalities involving generalized conformable integrals can be found in the work of Nisar et al. [42,43]. Recently, Tassaddiq et al. [63] proposed certain inequalities for the weighted and extended Chebyshev functionals by using fractional conformable integrals. Nisar et al. [40] presented some new classes of inequalities for an n (n ∈ N) family of positive continuous and decreasing functions via generalized conformable fractional integrals. Nisar et al. [41] established generalized fractional integral inequalities via the Marichev-Saigo-Maeda (MSM) fractional integral operators. Rahman et al. [54] recently investigated Grüss-type inequalities for generalized k-fractional conformable integrals. Minkowski's inequalities, fractional Hadamard proportional integral inequalities, and fractional proportional inequalities for convex functions by employing fractional proportional integrals can be found in [46][47][48][49][50][51][52][53]. In addition, various applications of fractional calculus can be found in [1, 17, 18, 28-32, 62, 64].
The paper is organized as follows. Some auxiliary results are presented in Sect. 2. In Sect. 3, we present double-weighted fractional integral inequalities for the Chebyshev functionals. In Sect. 4, we retrieve several particular cases of the results. A concluding remark is given in Sect. 5.

Some double-weighted generalized fractional integral inequalities
In this section, we present some double-weighted generalized fractional integral inequalities. We start by proving the following lemma.

Lemma 3.1 Let Ψ be a measurable increasing positive function on
Proof Suppose that U : [x 1 , Consequently, it follows that Utilizing the condition Applying (13) to the particular case U (x) = x, we can write The latter by (9) gives which completes the proof.
Based on Lemma 3.1, we prove the following theorem.
Proof By employing definition (9) and Lemma 3.1 we obtain Consequently, it follows that By the Cauchy-Schwarz inequality [11] we get Ψ Hence using (13) and (15), we conclude the proof.
Proof Considering the left-hand side of (16), we have Ψ Applying the Cauchy-Schwarz inequality [11] to this inequality, we get Ψ In view of (9), we get the desired proof of (16).
Proof Considering the left-hand side of (17), we have Ψ Hence taking (9) into account, we complete the proof of (17).
, we obtain the desired result.
Proof Considering the left-hand side of (18), we have Ψ Hence by (9) we complete the proof.