Some Grüss-type results via Pompeiu’s-like inequalities

In this paper, some Grüss-type results via Pompeiu’s-like inequalities are proved.

For two Lebesgue integrable functions f, g : [a, b] → R, consider theČebyšev functional: Grüss [10] showed that The constant 1 4 is best possible in (1.3) in the sense that it cannot be replaced by a smaller quantity. Another, however less known, result, though it was obtained byČebyšev [7], states that provided that f , g exist and are continuous on [a, b] and f ∞ = sup t∈ [a,b] f (t) . The constant 1 12 cannot be improved in the general case.
TheČebyšev inequality (1.6) also holds if f, g : [a, b] → R are assumed to be absolutely continuous and A mixture between Grüss' result (1.4) andČebyšev's one (1.6) is the following inequality obtained by Ostrowski [15]: provided that f is Lebesgue integrable and satisfies (1.5), while g is absolutely continuous and g ∈ L ∞ [a, b] . The constant 1 8 is best possible in (1.7). The case of Euclidean norms of the derivative was considered by Lupaş [12], in which he proved that provided that f, g are absolutely continuous and f , g ∈ L 2 [a, b] . The constant 1 π 2 is the best possible. Recently, Cerone and Dragomir [3] have proved the following results: where p > 1 and 1 p + 1 q = 1 or p = 1 and q = ∞, and . Notice that for q = ∞, p = 1 in (1.9), we obtain (1.11) and, if g satisfies (1.5), then (1.12) The inequality between the first and the last term in (1.12) has been obtained by Cheng and Sun [8]. However, the sharpness of the constant 1 2 , a generalization for the abstract Lebesgue integral and the discrete version of it have been obtained in [4].
In this paper, some Grüss-type results via Pompeiu's-like inequalities are proved.

Some Pompeiu's-type inequalities
We can generalize the above inequality for the larger class of functions that are absolutely continuous and complex valued as well as for other norms of the difference f − f .
Proof If f is absolutely continuous, then f / is absolutely continuous on the interval [a, b] that does not contain 0 and we get the following identity: We notice that the equality (2.3) was proved for the smaller class of differentiable function and in a different manner in [17].
Taking the modulus in (2.3), we have and utilizing Hölder's integral inequality we deduce 5) and the inequality (2.2) is proved.

Remark 2.2
The first inequality in (2.1) also holds in the same form for 0 > b > a.

Some Grüss-type inequalities
We have the following result of Grüss type.
The constant 1 12 is best possible.
Proof From the first inequality in (2.1), we have Utilizing the inequality (3.2), we deduce the desired result (3.1). Now, assume that the inequality (3.1) holds with a constant B > 0 instead of 1 12 , i.e., and f − f ∞ = g − g ∞ = 1 and by (3.3) we get B ≥ 1 12 , which proves the sharpness of the constant.
The following result for the complementary ( p, q)-norms, with p, q > 1 and 1 p + 1 q = 1, holds.

Theorem 3.2 Let f, g : [a, b] → C be absolutely continuous functions on the interval
We have the bounds a, b) where, for r > 1, Proof From the second inequality in (2.1), we have If we multiply these inequalities and integrate, then we get (3.5) Utilizing Hölder's integral inequality for double integrals, we have for p, q > 1 and 1 p + 1 q = 1. Utilizing Cauchy-Bunyakowsky-Schwarz integral inequality for double integrals, we have Therefore, and, similarly,

Remark 3.3 The double integral
can be computed exactly by iterating the integrals. However, the final form is too complicated to be stated here.
The Euclidian norms case is as follows: Proof From the second inequality in (2.1), we have If we multiply these inequalities and integrate, then we get Making use of the inequality (3.8), we deduce the desired result (3.7).

Remark 3.5
It is an open question to the author if 1 9 is best possible in (3.7). Theorem 3.6 Let f, g : [a, b] → C be absolutely continuous functions on the interval [a, b] with b > a > 0. Then, Proof From the third inequality in (2.1), we have which together with (3.10) produces the desired inequality (3.9).

Some related results
The following result holds.
We have Integrating this inequality on [a, b] 2 , we get