1. Introduction
The classical form of Grüss inequality, first published by G. Grüss
in 1935, gives an estimate of the difference between the integral of the
product and the product of the integrals of two functions. In recent years,
several bounds for the Cebysev functional in various cases including
convexity assumptions for the functions involved are proved. In the
subsequent years, many variants of these inequalities appeared in the
literature (see, [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12]).
In 1935, G. Grüss [
4]
proved the following inequality:
provided that and are two integrable function on satisfying
the condition
The constant is best possible.
In 1882, P. L. Čebyšev [
13] gave the following inequality:
where are absolutely continuous function,
whose first derivatives and are bounded,
and denotes the norm in defined as
In [
14], Beesack
et al. have proved the following Cebysev inequality
for absolutely continuous functions whose first derivatives belong to
spaces.
where and
In this paper, some inequalities related to Chebyshev’s functional are
proved. We give our results in the case of differentiable functions whose
derivatives and theirself belong to
2. Main results
Theorem 2.1.
Let be an absolutely continuous function on
so that and are convex on
- If , then we have
- If , then we have
- If , then we have
Proof.
For any we write
and so
Let’s rewrite (6) as follows
Multiplying (6) by and (7) by , adding the
resulting identities, and integrate over , and divide
by we have
and rewriting we get
(1) Thus, using the properties of modulus and the convexity of and , we have
for and the inequality (3) is proved.
(2) As above, we rewrite
Using the Hölder’s inequality for , we
have
and the inequality (4) is proved.
(3) We consider the inequality (9) that
and the inequality (5) is proved.
Theorem 2.2.
Let be an absolutely continuous function on
so that and with are convex on
- If , then we have
- If , then we have
- If , then we have
Proof.
By Hölder’s inequality and using the convexity of we get
From (8) and using the properties of modulus and the convexity of and , we have
for
(1) If we take , then from (
13), we have
for any the inequality is proved.
(2) If , then from (13) and by Hölder’s inequality we have
which is proved the inequality (11).
(3) If , then from (13)
and by Hölder’s inequality we also have
which is proved the inequality (12).
Competing interests
The authors declare that they have no competing interests.