On more general inequalities for weighted generalized proportional Hadamard fractional integral operator with applications

Fractional calculus has been the target of the work of many mathematicians for more than a century. Some of these investigations are of inequalities and fractional integral operators. In this article, a novel fractional operator which is known as weighted generalized proportional Hadamard fractional operator with unknown attribute weight is proposed. First, a fractional formulation is constructed, which covers a subjective list of operators. With the aid of the above mentioned operators, numerous notable versions of Pólya-Szegö, Chebyshev and certain related variants are established. Meanwhile, new outcomes are introduced and new theorems are exhibited. Taking into account the novel generalizations, our consequences have a potential association with the previous results. Furthermore, we demonstrate the applications of new operator with numerous integral inequalities by inducing assumptions on weight function $ and proportionality index φ. It is hoped that this research demonstrates that the suggested technique is efficient, computationally, very user-friendly and accurate.

Various notable generalized fractional integral operators such as the Riemann-Liouville, Hadamard, Caputo, Marichev-Saigo-Maeda, Riez, the Gaussian hypergeometric operators and so on, are helpful for researchers to recognize real world phenomena. Therefore, the Caputo, Riemann-Liouville and Hadamard were the most used fractional operators having singular kernels. It is remarkable that all the above mentioned operators are the particular cases of the operators investigated by Jarad et al. [27]. The utilities are currently working on weighted generalized fractional operators. Inspired by the consequences in the above mentioned papers, we introduce a new weighted framework of generalized proportional Hadamard fractional integral operator. Also, some new characteristics of the aforesaid operator are apprehended to explore new ideas, amplify the fractional operators and acquire fractional integral inequalities via generalized fractional operators (see Remark 2 below).
Our intention is to establish a more general form for the most appealing and noteworthy Pólya-Szegö-Chebyshev type inequalities [36,37] and certain related variants via weighted generalized proportional Hadamard fractional integral that could be increasingly practicable and, also, more appropriate than the existing ones.
In 1882, Chebyshev pondered the noted result [36]: The intensively studied Grüss inequality [52] for two integrable functionsf andg on [η 1 , η 2 ] is presented as follows: where the integrable functionsf andg satisfy q ≤f ≤ Q and s ≤f ≤ S for all x ∈ [η 1 , η 2 ] and for some q, s, Q, S ∈ R. The Pólya-Szegö type inequality [37] can be stated as follows: (1. 3) The constant 1 4 is best feasible in (1.3) make the experience it cannot get replaced by a smaller constant. With the aid of the Pólya-Szegö inequality, Dragomir and Diamond [53] derived the inequality  [56][57][58][59][60][61][62][63] and the references cited therein. The motivation for this paper is twofold. First, we introduce a novel framework named weighted generalized proportional Hadamard fractional integral operator, then current operator employed to on the Pólya-Szegö-Chebyshev and certain related inequalities for exploring the analogous versions of (1.1) and (1.3). The study is enriched by giving remarkable cases of our results which are not computed yet. Interestingly, particular cases are designed for Hadamard fractional integral, generalized proportional Hadamard fractional integral and weighted Hadamard fractional integral inequalities. It is worth mentioning that these operators have the ability to recapture several generalizations in the literature by considering suitable assumptions of and ϕ.

Prelude
This section demonstrates some essential preliminaries, definitions and fractional operators which will be utilized in this paper.
and f χ p = ess sup Now, we show a novel fractional integral operator which is known as the weighted generalized proportional Hadamard fractional integral operator as follows.
Remark 2. Some particular fractional operators are the special cases of (2.5) and (2.6). I. Setting (x) = 1 in Definition 2.2, then we get the generalized proportional Hadamard fractional operator introduced by Rahman et al. [62] stated as follows: II. Setting ϕ = 1 in Definition 2.2, then we get the weighted Hadamard fractional operators stated as follows: Proof. By means of given hypothesis, we obtaiñ (2.14) Thus, we have which imply that Here, taking product each side of the above inequality by the following term 1 and integrating the resulting inequality with respect to on [η 1 , x], we have (2.17) Multiplying both sides of the above equation by −1 (x) and employing Definition 2.2, we have Taking into account the arithmetic-geometric mean inequality, we have which leads to the inequality (2.13). This completes the proof.
Proof. By means of assumption (2.12), we have ≥ 0 and f ( ) which imply that Conducting product each side of the inequality (2.24) by υ 1 (φ)υ 2 (φ)g 2 (φ), we get Here, taking product each side of the above inequality by the following term and integrating the resulting inequality with respect to and φ on [η 1 , x]. Then, multiplying both sides of the inequality by −2 (x) and employing Definition 2.2, we obtain By employing the arithmetic-geometric mean inequality, we have which leads to the desired inequality (2.22). Hence the proof is complete.
Proof. By means of assumption (2.12), we have Multiplying both sides of the above equation by −1 (x) and employing of Definition 2.2, we have By similar argument, we have Multiplying both sides of the above equation by −1 (x) and employing Definition 2.2, we have Taking product of the inequalities (2.31) and (2.33) side by side, then we obtain the desired inequality (2.29).
Our next result is the Chebyshev type integral inequality within the weighted generalized proportional Hadamard fractional integral operator defined in (2.3), with the aid of Pólya-Szegö type inequality established in Theorem 2.3.
Proof. The inequalities (c 3 ) − (c 6 ) can be deduced by utilizing Lemma 2.7 and the following assumptions: IV. Setting ϕ = 1, then we get a new result for generalized proportional Hadamard fractional integral operator.

Proof. In view of Definition 2.2 and applying modulus property that
for φ > 1. By the virtue of the noted Hölder inequality, we have Substituting ν = ln x φ . Then elaborated computations represents

Applications
In the sequel we demonstrate a new methodology for establishing the four bounded mappings and employ them to show certain bounds of Chebyshev type weighted generalized proportional Hadamard fractional integral inequalities of two unknown mappings. Consider a unit step function χ be defined as and assuming a Heaviside unit step function χ η 1 (x) defined by The main characteristic of the unit step function are its frequent use in the differential equations and piece-wise continuous functions when sum of pieces defined by the series of functions. Assume that a piece-wise continuous function υ 1 ( ) defined on [η 1 , T ] can be presented a follows: where h 0 , h j ∈ R( j = 0, 1, ..., q) and η 1 = x 0 < x 1 < x 2 < ... < x i < x q = T . Analogously, we define the mappings υ 2 , υ 3 and υ 4 as follows where r 0 = R 0 = H 0 = 0 and r j , R j , H j ∈ R ( j = 0, 1, ..., q).
Remark 9. The accuracy of the approximated estimates (4.5) and (4.8) depends on the value of q ∈ N.

Conclusions
This paper proposes a new generalized fractional integral operator. The novel investigation is used to generate novel weighted fractional operators in the Hadamard and generalized proportional Hadamard fractional operator, which effectively alleviates the adverse effect of weight function and proportionality index ϕ. Utilizing the weighted generalized proportional Hadamard fractional operator technique, we derived the analogous versions of the weighted Pólya-Szegö-Chebyshev and certain associated type inequalities that improve the accuracy and efficiency of the proposed technique. Contemplating the Remark 2, several existing results can be identified in the literature. It is important to note that our generalizations are refinements of the results obtained by [69]. Some innovative particular cases constructed by this method are tested and analyzed for statistical theory, fractional Schrödinger equation [35]. The results show that the method proposed in this paper can stably and efficiently generate integral inequalities for convexity with better operators' performance, thus providing a reliable guarantee for its application in control theory [67].