Certain new proportional and Hadamard proportional fractional integral inequalities

The main goal of this paper is estimating certain new fractional integral inequalities for the extended Chebyshev functional in the sense of synchronous functions by employing proportional fractional integral (PFI) and Hadamard proportional fractional integral. We establish certain inequalities concerning one- and two-parameter proportional and Hadamard proportional fractional integrals. We also discuss certain particular cases.


Introduction
The integral inequalities play a major role in the field of differential equations and applied mathematics. Applications of integral inequalities are found in applied sciences, such as statistical problems, transform theory, numerical quadrature, and probability.
In the last few years, many researchers have established various types of integral inequalities by employing different approaches. The interested readers are suggested to see [4,5,8,14,15,17,18]. In [28,[39][40][41] the researchers established different kinds of integral inequalities by employing various types of fractional integrals.
In the last few years, the field of fractional calculus has been extensively studied due to wide applications in diverse domains. Several different kinds of fractional integral and derivative operators have been investigated. We refer the readers to [1-3, 6, 7, 11, 21, 24].
In [20,22] the authors introduced the idea of fractional conformable integral operators.
We consider the following extended Chebyshev functional: The functions U and V are said to be asynchronous on [r, s] if the inequality is reversed, that is, If the functions U and V are synchronous on [r, s], then T(U , V, μ, ν) ≥ 0. For more detail, see Kuang [23] and Mitrinovic [26]. The Chebyshev functional (1) If the functions U and V are asynchronous on [r, s], then The researchers studied the functional T(U , V, μ) and established several extensions and generalizations, which can be found in [9,10,16,25].

Preliminaries
In this section, we present some well-known definitions and mathematical preliminaries of fractional calculus. and where (κ) is the classic gamma function.
In this paper, we consider the following one-sided PFI-operator.

Definition 2.4
The one-sided PFI is defined by where κ > 0 is the order of PFI, and ω ∈ (0, 1] is the proportionality index.

Definition 2.5
The left-sided Hadamard fractional integral of order κ > 0 is defined by Definition 2.6 The right-sided Hadamard fractional integral of order κ > 0 is defined by

Definition 2.7
The one-sided Hadamard fractional integral of order κ > 0 is defined by Rahman et al. [32] recently presented the following generalized Hadamard proportional fractional integrals.

Definition 2.9
The right-sided Hadamard proportional fractional integral is defined by
The following results can be easily proved. and (the semigroup property). Remark 2.3 Setting ω = 1, (13) reduces to (see [38]) The paper is organized as follows. In Sect. 3, we present two integral inequalities for the extended Chebyshev functional. The first result concerns one-parameter PFI, and the second one deals with two-parameter PFI. In Sect. 4, we establish integral inequalities for the extended Chebyshev functional by employing the Hadamard proportional fractional integral.
Remark 3.5 Inequalities (20) and (27) will be reversed in the following cases: