Chebyshev-Type Inequalities Involving ()-Proportional Fractional Integral Operators
Expanding the analytical aspect of mathematics enables researchers to study more cosmic phenomena, especially with regard to the applied sciences related to fractional calculus. In the present paper, we establish some Chebyshev-type inequalities in the case synchronous functions. In order to achieve our goals, we use -proportional fractional integral operators. Moreover, we present some special cases.
The generalization of differentiation and integration of integer order to noninteger order is the subject of study in Fractional Calculus (FC). The theory of FC has become a subject of great interest with many applications, such as viscoelasticity , bioengineering , and control theory . There are several definitions of fractional operators appearing in the literature, ranging from the most well-known Riemann-Liouville and Caputo to others such as Hadamard, Erdèlyi-Kober, Riesz, and Prabhakar [4, 5]. Miller and Ross  and Samko et al.  have written a research monograph that goes into great detail about fractional operators, including their properties and applications.
In the last few decades, the development of FC theory has been taken into consideration by many researchers. Since then, researchers have primarily focused on the generalizations of fractional operators by exhibiting different kernels, which has made a significant contribution to the development of FC. In particular, Jarad et al.  has contributed significantly to the study of FC by introducing generalized proportional fractional operators through exponential kernel, while in , the authors extended this work and defined the proportional fractional operator of a function with respect to the another function. Furthermore, in , Aljaaidi et al. extended the studies on proportional fractional operators and proposed new fractional operators known as -proportional fractional operators.
On the other hand, due to the importance of fractional operators, many different types of inequalities have been established by researchers using various fractional operators. For instance, Set et al.  established Pólya-Szegö-type inequalities by means of a generalized proportional Hadamard fractional operator. Rashid et al.  presented reverse Minkowski’s type inequalities by using generalized proportional fractional operators. In , authors studied Pólya-Szegö-type inequalities and Grüss-type inequalities with the help of -proportional fractional operators. In recent years, many researchers have been working in the direction of estimating the fractional version of various inequalities [13–19] and the references given there.
In 1882, Chebyshev introduced the following functional (see ):
Let and be an integrable function on the interval . Then,
If and are synchronous, that is,
Let and be the integrable function on and be the positive and integrable function on Then,
Many researchers have given considerable attention to both functionals, and a number of inequalities and a number of extensions, generalizations, and variants have appeared in the literature over the past few decades (see [21–26]).
Furthermore, taking as a positive and integrable function on , the extended Chebyshev functional is given as follows (see ):
In the literature, many authors provide the results about the Chebyshev-type inequality for various fractional operators. In 2009, Belarbi and Dahmani in  presented Chebyshev-type inequalities related to functional (1) by employing the Riemann-Liouville definition. We also note that the Chebyshev-type inequalities for the extended Chebyshev functional established by Dahmani in  used the same operator. In 2016, Sarikaya et al. in  introduced a new definition -Riemann-Liouville fractional and obtained Chebyshev-type inequalities for it in the case of functional (1). Chebyshev-type inequalities were studied by Set et al. in  via extended generalized fractional operators. In , authors used the generalized Katugampola operators to establish integral inequalities for Chebyshev and extended Chebyshev functionals. In the literature, a number of mathematicians have devoted their efforts to study Chebyshev-type inequalities using various fractional operators [33–38].
Motivated by the above works, our aim in this paper is to establish some new Chebyshev-type inequalities. We use -proportional fractional operators to establish our main results. The rest of the paper is organized as follows: In Section , we give some preliminaries and definitions which will be useful in the sequel. In Section , we prove some Chebyshev-type inequalities related to functional (1) and the extended Chebyshev functional. In Section , we give the concluding remark.
In this section, we give some basic definitions, preliminaries which are useful for our subsequent discussions:
Definition 1 (see ). Let Let be an integrable function on . Then, generalized proportional fractional integral of of order is given by where is the Gamma function.
Definition 2 (see ). Let Let be an integrable function and be a continuous increasing function on . Then, generalized proportional fractional integral of with respect to of order is given by where , is the Gamma function.
Definition 3 (see ). Let Let be an integrable function and be a continuous increasing function on . Then, the -proportional fractional integral of with respect to of order is given by where , , is the -Gamma function.
3. Chebyshev-Type Inequalities
This section deals with the Chebyshev-type integral inequalities for synchronous functions.
Theorem 4. Let . Assume that and are integrable and synchronous functions on Then
Proof. Since and are synchronous functions on for all , we have Then, Now, on both sides of (11), taking multiplication by where ; then, integrating with respect to from 0 to we obtain Consequently, Multiplying both the sides of (13) by where ; then, integrating with respect to from 0 to we get Hence,
Theorem 5. Let and . Assume that and are integrable and synchronous functions on Then,
Proof. Since and are synchronous on , using the same argument as in Theorem (9), we obtain Now, on both sides of (17), taking multiplication by where ; then, integrating with respect to from 0 to we have From above, In the next result, we prove some integral inequalities for synchronous function related to the extended Chebyshev functional in the case of one fractional parameters. First, we prove the following lemma in order to use it to prove our next theorem.
Lemma 6. Let . Assume that and be integrable and synchronous functions on . Suppose , : . Then,
Proof. Since and are synchronous functions on for all , we have Then, On both sides of (22), taking multiplication by where ; then, integrating with respect to from 0 to we get Multiplying both sides of (23) by where ; then, integrating with respect to from 0 to we get
Theorem 7. Let . Assume that and are integrable and synchronous functions on . Suppose : . Then, for ,
Proof. Substituting in (20), then multiplying by we get
Putting in (20), then multiplying by we obtain
Now putting in (20), then multiplying by we have
Adding (26), (27), and (28), we get (25).
In the next result, we prove some integral inequalities for the synchronous function related to the extended Chebyshev functional using two fractional parameters. First, we prove the following lemma in order to use it to prove our next theorem.
Lemma 8. Let Assume that and are integrable and synchronous functions on and , : . Then,
Theorem 9. Let . Assume that and are integrable and synchronous functions on . Suppose : . Then,