Generalized Integral Inequalities of Chebyshev Type

: In this paper, we present a number of Chebyshev type inequalities involving generalized integral operators, essentially motivated by the earlier works and their applications in diverse research subjects.


Introduction
Many integral inequalities of various types have been presented in the literature. Among them, we choose to recall the following Chebyshev inequality (see [1]): where f and g are two integrable and synchronous functions on [a, b], a < b, a, b ∈ R. Here, two functions f and g are called synchronous on [a, b] if In the case that we have − f and g (or similarly f and −g) the sense of the previous inequality is the opposite.
Inequality (1) 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 inequality (1), these are called Chebyshev type inequalities (see, e.g., [2,3] or [4]).
We give the definition of a general fractional integral. We assume that the reader is familiar with the classic definition of the Riemann integral, so we will not present it. Throughout the paper we will suppose that the positive integral operator kernel T : I → (0, ∞) defined below is an absolutely continuous function on interval I ⊆ R. Definition 1. Let I be an interval I ⊆ R and a, b ∈ I. The generalized integral operators J T,a + and J T,b − , called respectively, right and left, are defined for every locally integrable function f on I as follows: dt, x > a.
Note that in special cases, J T,a + and J T,b − are equal to the following integrals: We say that f belongs to the function space It is easy to see that the case of the J T operators defined above contains, as particular cases, the integral operators obtained from conformable and non-conformable local derivatives. For details about the Riemann-Liouville fractional integrals (left-sided) of a function f of order α ∈ C with Re(α) > 0 the reader can consult [5,6]. In [7], Belarbi and Dahmani established some theorems related to the Chebyshev inequality involving Riemann-Liouville fractional integral operator. Recently, some new integral inequalities involving this fractional integral operator have appeared in the literature, see, e.g., [8][9][10][11][12][13][14][15][16][17][18][19].
Taking into account the previous research results and the generalized integral operator, we will obtain some Chebyshev type inequalities, which contain many of the inequalities reported in the literature as particular cases.

Main Results
Theorem 1. Let f and g be two functions from L + T [a, b] which are synchronous on [a, b]. Then where .
Proof. Since f and g are synchronous on [a, b], we have Multiplying both sides by 1 T(u−a) yields Integrating both sides of the resulting inequality with respect to the variable u from a to b, gives us b a f (u)g(u) From this, we have After multiplying the inequality by 1 T(v−a) and integrating with respect to v between a and b, we get and we have got (2).

Remark 1. Similar calculations as above shows that for any f
Remark 2. If we take T ≡ 1 in Theorem 1 (or in Remark 1), then inequality (2) (or (4)) reduces to the classic inequality (1) of Chebyshev.

Theorem 2. Let f and g be two functions from L
Then where Proof. Writing T 1 in place of T and τ 1 in place of τ in (3) and then multiplying both sides by .
Integrating both sides of the resulting inequality with respect to the variable v between a and b gives us (6).

Remark 4.
In case of T 1 = T 2 , we obtain Theorem 1.
Proof. We prove this theorem by induction on n ∈ N. For n = 1, (7) trivially holds. For n = 2, (7) immediately comes from (2), since f 1 and f 2 are synchronous on [a, b]. Now assume that the inequality (7) is true for some n ∈ N. Let f := ∏ n i=1 f i and g := f n+1 . Observe that f and g are increasing on [a, b], therefore (2) and the induction hypothesis for n yields From the equality and this equality together with (9)-(12) implies the required result.

Remark 10.
The results obtained in this work can be extended if we consider instead of f and g, − f and g or f and −g, in the notion of synchronous functions, in which case the direction of the inequalities changes .

Conclusions
In this work, we have obtained the Chebyshev inequality from Theorem 1 within the framework of generalized integrals. In addition to the observations made, which prove the strength of our results, we would like to present a couple of variants of the classic Chebyshev inequality.
If we take kernel T = t α , α < 1, then we get