A note on Chebyshev inequality via k -generalized fractional integrals

In this paper, by using the k -generalized fractional integrals, we establish certain integral inequalities for the Chebyshev functional in case of synchronous function. The obtained inequalities generalize several known integral inequalities.


Introduction
One of the most developed mathematical areas in recent years is that of integral inequalities, particularly involving various fractional and generalized integral operators; for example, see [1, 5-8, 10, 15, 17, 18, 21, 25, 31, 43, 44]. Recently, the generalized k-proportional fractional integral operators with general kernel were defined, which contain many of the known fractional operators. In order to give detail of this work, we need to first present some preliminary results.
Throughout this paper, we use the functions Γ (see [33,36,46,47]) and Γ k (see [13]) defined as follows: It is clear that if k → 1 we have Γ k (z) → Γ(z), Γ k (z) = (k) z k −1 Γ z k and Γ k (z + k) = zΓ k (z). Also, we define the k-beta function as follows Notice that The fractional integral operator of Riemann-Liouville are being extended and generalized in various ways. One of such ways are presented in this paper. From the point of view of differential operators and by manipulating simple algebraic identities, one can follow the idea of fractional differential operators of the Riemann-Liouville or Caputo type. From the simple facts α = 1 + α − 1 and α = α − 1 + 1 we have and respectively. Next, we present several definitions of fractional integrals, some very recent (with 0 ≤ a 1 < τ < a 2 ≤ ∞). One of the first operators that can be called fractional is the Riemann-Liouville fractional derivative of order α ∈ C, with Re(α) > 0, defined as follows (see [16]). Definition 1.1. Let a 1 < a 2 and f ∈ L 1 ((a 1 , a 2 ); R). The right and left side Riemann-Liouville fractional integrals of order α, with Re(α) > 0, are defined, respectively, by with t ∈ (a 1 , a 2 ).
Other definitions of fractional integral operators are the following ones.
In [22], the author introduced new fractional integral operators, called the Katugampola fractional integrals, in the following way. Definition 1.3. Let 0 < a 1 < a 2 , f : [a 1 , a 2 ] → R be an integrable function, and α ∈ (0, 1), ρ > 0 be two fixed real numbers. The right and left side Katugampola fractional integrals of order α are defined, respectively, by The left-sided and right-sided Riemann-Liouville k-fractional integrals are given in [28].
. Then the Riemann-Liouville k-fractional integrals of order α ∈ C, (α) > 0 and k > 0 are given by the expressions: A more general definition of the Riemann-Liouville fractional integrals is given in [24]. Definition 1.5. Let f : [a 1 , a 2 ] → R be an integrable function. Also, let g be an increasing and positive function on (a 1 , a 2 ] with a continuous derivative g on (a 1 , a 2 ). The left and right sided fractional integrals of a function f with respect to another function g on [a 1 , a 2 ] of order α ∈ C, (α) > 0, are expressed by: A k-fractional analogue of Definition 1.5 is given in the following (see [4,26,37]). Definition 1.6. Let f : [a 1 , a 2 ] → R be an integrable function. Also, let g be an increasing and positive function on (a 1 , a 2 ] with a continuous derivative g on (a 1 , a 2 ). The left and right sided k-fractional integrals of a function f with respect to another function g on [a 1 , a 2 ] of order α ∈ C, (α) > 0 and k > 0 are expressed by: Next, we have the definition of the generalized proportional fractional (GPF) integral operator (see [38]). Definition 1.7. Let U ∈ X q Ψ (0, +∞), 0 < a 1 < a 2 , and there be an increasing and positive monotone function Ψ defined on [0, +∞) having continuous derivative Ψ on [0, +∞) with Ψ(0) = 0. Then the left-sided and right-sided GPF-integral operator of a function U in the sense of another function Ψ of order η are stated as: where the proportionality index ς ∈ (0, 1], η ∈ C, Re(η) > 0, and Γ is the gamma function.
The functional space on which we develop our work is the following.
and for the case q = +∞ We are now in a position to define the generalized integral operators that we use in our work (see [30]).

Definition 1.9.
Let h ∈ X q F (0, +∞) and F be a continuous and positive function on [0, +∞) with F (0) = 0. The right and left side generalized k-proportional fractional integral operators with general kernel of order γ of h are defined, respectively, by and where the proportionality index λ ∈ (0, 1), γ ∈ C, Re(γ) > 0, χ ∈ (a 1 , a 2 ), Of course there are other integral fractional operators and their variations can be considered, however we do not discuss them here. Remark 1.1. Next, we will show how many integral operators are particular cases of (1) and (2).
2. Under the above conditions, if k = 1 then, from Definition 1.9, the k-fractional operators defined in [28] are obtained.
8. If we Choose λ = 1, F (s) = g (s), k = 1 and then we obtain the integral operator defined in [38], known as GFP and is given in Definition 1.7.

Results
One of the best known integral inequalities is Chebyshev's inequality (see [11]), which establishes relationships between the integral of the product of two functions and the product of their integrals. This inequality is stated in the framework of the classical Riemann integral: where f and g are two integrable and synchronous functions on [a, b], a < b, a, b ∈ R. Inequality (3) has many applications in diverse research subjects such as numerical quadrature, transform theory, probability, existence of solutions of differential equations and statistical problems. Many authors have investigated generalizations of the Chebyshev's inequality (3), which are called Chebyshev type inequalities (for example, see [2,9,12,14,20,27,29,32,34,[41][42][43]45]). This inequality is generalized also in the present paper by using the generalized operator of Definition 1.9, which contain many inequalities reported in the literature as particular cases.
In Chebyshev's work cited above, the following functional is presented, which has been the subject of researches: where f and g are two integrable functions which are synchronous on [a, b].
Proof. As f and g are synchronous functions on [0, ∞), it follows that Multiplying both sides of (5) by 1 λ γ kΓ k (γ) and integrating the result with respect to u over (a 1 , χ), we get Multiplying both sides of (6) by 1 λ γ kΓ k (γ) then integrating the resulting inequality with respect to v over (a 1 , χ), we obtain that is, Reordering it and taking into account the definition of C γ k λ (f, g), the desired inequality is obtained. This completes the proof. . Then for all χ > a 1 ≥ 0, λ ∈ (0, 1) and γ, δ ∈ C with Re(γ) > 0, Re(δ) > 0 it holds that is the generalization of the Functional T (f, g) for γ and δ.
Proof. Multiplying both sides of (6) by 1 λ δ kΓ k (δ) then integrating the resulting inequality with respect to v over (a 1 , χ), we obtain Dividing both sides of (8) by .
The previous results can be extended if we consider a certain positive "weight" function h. Theorem 2.4. Let f and g be two synchronous functions on [0, ∞) and h ≥ 0. Then for all χ > a 1 ≥ 0, λ > 0 and γ ∈ C with Re(γ) > 0, the following inequality holds: Proof. Since h ≥ 0 and the functions f and g are synchronous on [0, ∞), we obtain Multiplying both sides of (9) by 1 λ γ kΓ k (γ) then integrating the resulting inequality with respect to u over (a 1 , χ), we obtain We can write as Multiplying both sides of (10) by 1 λ γ kΓ k (γ) then integrating the resulting inequality with respect to v over (a 1 , χ), we obtain That is, Then for all χ > a 1 ≥ 0, λ ∈ (0, 1), γ ∈ C with Re(γ) > 0 and δ ∈ C with Re(δ) > 0, the following inequality holds: Proof. Multiplying both sides of (10) by 1 λ δ kΓ k (δ) then integrating the resulting inequality with respect to v over (a 1 , χ), we obtain

Remark 2.4. For h = 1, Theorem 2.5 gives Theorem 2.2.
More precise results can be obtained, if in the previous Theorem we impose additional conditions on the function h. Theorem 2.6. Let f, g and h be three monotonic functions defined on [0, ∞) satisfying the following inequality Then, for all χ > a 1 ≥ 0, λ ∈ (0, 1), γ ∈ C with Re(γ) > 0 and δ ∈ C with Re(δ) > 0, it holds that Proof. As in the proof of Theorem 2.5, if we multiply both sides of (10) by then integrate the resulting inequality with respect to v over (a 1 , χ), we obtain the desired inequality.
An inequality involving the square of the functions f and g can be stated as follows.
we have Multiplying both sides of (13) by 1 λ γ kΓ k (γ) then integrating the resulting inequality with respect to u and v over (a 1 , χ), we obtain (11).
On the other hand, since then by using the same arguments as before, we have (12).
On the other hand, since (f (u)g(v) − f (v)g(u)) 2 ≥ 0, then by using the same arguments as before, we have (15). This complete the proof.

Conclusion
In this paper, we present a generalized formulation of the Riemann-Liouville fractional integral, which contains as particular cases many of the integral operators reported in the literature. We present several integral inequalities that generalize several known inequalities. We highlight the strength of Definition 1.9 by pointing out the following fact. If we consider the kernel F (χ, s) = χ 1−s and G ≡ 1, we obtain a variant of the (k, s)-Riemann-Liouville fractional integral defined in [40]: This opens up wide possibilities of obtaining new integral inequalities.