On some inequalities relative to the Pompeiu–Chebyshev functional

In this paper we study the utility of the functional Pompeiu–Chebyshev in some inequalities. Some results obtained by Alomari will be generalized regarding inequalities with Pompeiu–Chebyshev type functionals, in which linear and positive functionals intervene. We investigate some new inequalities of Grüss type using Pompeiu’s mean value theorem. Improvement of known inequalities is also given.


Introduction
If f , g are integrable functions on [a, b], then the functional defined by is known as functional Chebyshev, with multiple applications in numerical analysis and probability theory (see [3]).
The following theorem combines a series of results regarding the bounds for this functional.
In 2005, Pachpatte (see [8]) introduced the following functional. If f , g : [a, b] → R are two differentiable functions on (a, b), then and proved the following result.
Dragomir (see [4]) studied the Pompeiu-Chebyshev functional and changed it as follows: The following result, obtained by Dragomir in [4], will be used in some demonstrations included in this paper.

Lemma 1.4 Let f : [a, b] → R be an absolutely continuous function on
where p, q > 1 with 1 p + 1 q = 1.
In [1], Alomari studied and generalized some inequalities related to the Pompeiu-Chebyshev functional.
The purpose of this paper is to generalize the results of Alomari considering the Pompeiu-Chebyshev functional in which linear and positive functionals intervene.

Main results
In the following we denote by F a set of linear functions defined on the interval I = [a, b]. We will assume that the set F contains the constant and polynomial functions, and we suppose that, if f , g ∈ F , then f · g ∈ F .
Definition 2.1 Let A, B : F → R be two linear and positive functionals.
We call the functional P A,B (f , g) a Pompeiu-Chebyshev functional.
Remark 2.2 For any two linear and positive functionals A, B : F → R, we have then the functional P A,B (f , g) becomes the functional that was studied by Dragomir in [4].
Proof From Lemma 1.4 we have Next, by A x or B y we will understand that the functional A, respectively B, acts on the variable x, respectively y. It is easy to see that From relations (9) and (10) we get the following: Further we have Combining relations (11) and (12), we get (8).

Corollary 2.5 If we take
which is the inequality obtained by Dragomir in [4].
Definition 2.6 Let f , g ∈ F . The functions f and g are called synchronous (or similarly ordered) if for all x, y ∈ I, where I is the domain for f and g, we have and f and g are called asynchronous (or oppositely ordered) if for all x, y ∈ I we have Theorem 2.7 Let f , g ∈ F , where f , g : I → R such that 0 / ∈ I.
(i) If f e 1 and g e 1 are synchronous functions, then P A,B (f , g) ≥ 0. (ii) If f e 1 and g e 1 are asynchronous functions, then P A,B (f , g) ≤ 0.
Proof Since f e 1 and g e 1 are synchronous (asynchronous) functions, we have So, The following theorem shows a pre-Grüss inequality for the functional P A,B (f , g) (see [5]).
Constant 1 is the best possible.
Proof From the equality and from the CBS-inequality, we obtain From the above the conclusion is obtained.
We notice that for f (x) = g(x) = c · x -1, c ∈ R, fixed we obtain the equality in (16). We note that for

then the following inequality
holds.
Proof From the assumptions of the theorem we have Adding the last inequalities, we have In the same way we proceed for the function g, and we get From (18) we get we get the inequality from the conclusion.
dx, then we get the following inequality: Theorem 2.12 Let A, B : F → R be two linear and positive functionals.
, then the following inequality holds: Proof From the assumptions of the theorem we have Adding the last inequalities, we have In the same way we proceed for the function g, and we get From the above we get and then The last inequality is equivalent to the conclusion.
, then the following inequality holds.
Proof From (16) we get From (10) and (17) we have From relationships (22) and (23) we get the conclusion.
then we obtain the following inequality: Theorem 2.16 Let A, B : F → R be two linear and positive functionals.
, then the following inequality holds: Proof From (16) we get From (17) for f = g we have Replacing (27) in (26), we get the conclusion.
Proof It is easy to see, in these conditions, that we have From the above we obtain Applying the linear and positive functional A x B y and considering that the statement results. (22) where Proof Using the Cauchy-Schwarz inequality in (28), we have From the above we obtain Computing, we obtain So, we get where K f and K g are given in (30), respectively (31), which is the inequality from the conclusion.
A more general case is taken forward, which improves relationship (25).

Theorem 2.21
If there exist real numbers m g , M g such that m g ≤ g(x) ≤ M g , ∀x ∈ D, then where K g is given by (31).

Applications
In this section we investigate some new inequalities of Grüss type using Pompeiu's mean value theorem and the above results. Improvement of known inequalities is also given.
, then the following inequality holds.
Proof From Lemma 1.4 we have that and it follows that (29) we get the first result from [1, Th. 13].

then the following inequality
holds.
Proof From Lemma 1.4 we have where K g is given by (31).
Proof From inequality (8)  On the other hand, from inequality (29) we have Using the last two inequalities in (16), we get the conclusion. (37) we get the result from [1, Th. 14]. a, b]. Then we have the following inequality:

Theorem 3.7 Let
Proof From Lemma 1.4 we get Therefore, we have Using the last inequality in (39), we obtain (38).
dx, then inequality (39) becomes the following inequality: ]. Then we have the following inequality: Proof From Lemma 1.4 we get  a, b]. If there exist real numbers m g , M g such that m g ≤ g(x) ≤ M g , ∀x ∈ [a, b], then the following inequality holds: where K g is given by (31).
Proof From Lemma 1.4 we get Using inequality (29), we have where K g is given by (31). Substituting in (16) we get the desired result. Definition 3.13 Let a, b ∈ R, with a < b and f , g, h ∈ F , h : [a, b] → R + . The functional noted by P A,B (f , g; h), defined by is called Pompeiu-Chebyshev with respect to the function h functional.
Note that the functional can also be written in the following form: .
and f , g are called asynchronous with respect to a function h (h-asynchronous, oppositely The next result generalizes the inequalities from Theorem 2.7. From this and (43) we have from where we get the conclusion. The next theorem is a generalization of Theorem 2.9 and contains the pre-Grüss inequality.
Proof Using the CBS inequality in equality (43), we obtain and 1 2 A x B y h(x)g(y)h(y)g(x) 2 = P A,B (g, g; h).
From the above we get the conclusion. It is easy to see that the Pompeiu-Chebyshev functional with respect to the function h, P A,B (f , f ; h) represents the reverse of CBS-inequality. We have for α, β > 1, 1 Starting from this we can state the following results.
Proof From f h and g h are of p -H-Hölder type with H 1 , H 2 > 0, p, q ∈ (0, 1] we have By multiplying the last two inequalities, we obtain Using (43) in the last inequality P A,B (f , g; h), we obtain the conclusion.