Certain fractional conformable inequalities for the weighted and the extended Chebyshev functionals

*Correspondence: a.tassaddiq@mu.edu.sa 1Department of Basic Sciences and Humanities, College of Computer and Information Sciences, Majmaah University, Al Majmaah, Saudi Arabia Full list of author information is available at the end of the article Abstract The main aim of this present paper is to establish fractional conformable inequalities for the weighted and extended Chebyshev functionals. We present some special cases of our main result in terms of the Riemann–Liouville fractional integral operator and classical inequalities.


Introduction
Fractional calculus is the study of integrals and derivatives of arbitrary order which was a natural outgrowth of conventional definitions of calculus integral and derivative. In all areas of sciences, especially in mathematics, fractional calculus is a developing field with deep applications, though the idea was introduced more than three hundred years ago.
Many theories of mathematics applicable to the study of fractional calculus were emerging at the end of the 19th century.
In [7], the Chebyshev functional for two integrable functions f and g on [a, b] is defined as In [3,4,15,17], the applications and several inequalities related to (1) are found. In ( [10], also see [7]), the Chebyshev functional is defined by where f and g are integrable on [a, b] and h is a positive and integrable function on [a, b].

Preliminaries
In this section, we present the following well-known definitions from [20,23].
where Γ is the gamma function; for further details as regards gamma and related functions, see [45].

Definition 2.2
The fractional conformable integral β I α 0 of order β > 0, for a continuous function is defined by Clearly one can get 0 I α 0 f (τ ) = f (τ ) and In [23,35,37,38,40,43], one has studied fractional conformable integral operators and has established certain inequalities by employing the said fractional integral operators.

Main results
In this section, we establish certain fractional conformable inequalities for the weighted and the extended Chebyshev functionals.
By considering α = 1 in Theorem 3.1, we get the following well-known result of Dahmani et al. [12].
If we consider α = 1 in Theorem 3.2, then we get the following well-known result [12].

Concluding remarks
In this paper, we established certain fractional conformable inequalities related to the weighted and the extended Chebyshev functionals. The inequalities obtained in the present paper are more general than the existing classical inequalities cited therein. This work will reduce to the inequalities some Riemann-Liouville integral inequalities by taking α = 1, which have been presented earlier by [12]. Also, one can get the classical results by taking α = β = 1.