- Research Article
- Open Access
- Published:

# On Generalizations of Grüss Inequality in Inner Product Spaces and Applications

*Journal of Inequalities and Applications*
**volume 2010**, Article number: 167091 (2010)

## Abstract

Using a Kurepa's result for Gramians, we achieve refinements of well-known generalizations of Grüss inequality in inner product spaces. These results are further applied in to derive improvements of some published trapezoid-Grüss and Ostrowki-Grüss type inequalities. Refinements of the discrete version of Grüss inequality as well as a reverse of the Schwarz inequality are also given.

## 1. Introduction

Let be an orthonormal system of vectors in unitary space It is well known that for all , the following inequality holds [1, page 333]:

where is defined by

The equality in (1.1) holds if and only if is linearly dependent. Applying (1.1) on for by choosing , and , we immediately obtain the Pre-Grüss inequality as follows:

where and is the Chebyshev functional

Let , , , and be real numbers such that and for all Combining (1.3) with the following well-known inequality:

we obtain a premature Grüss inequality,

and the original Grüss inequality (see [2]),

Note that the discrete version of inequality (1.7) has the following form:

where are real numbers so that , for all , and , .

In [3, 4], Dragomir starting from inequality (1.2) proved the following Grüss type inequality in real or complex inner product spaces.

Let be a unit vector in If , , , and are complex numbers and are vectors in that satisfy the conditions,

then the following inequality holds:

The constant that appears at the right side of the inequality is optimal in the sense that it cannot be replaced by a smaller one.

Some generalizations and refinements of inequality (1.10) can be found in [1, 4–10] In this paper, we achieve an improvement of inequality (1.1) in the sense of subtracting a nonnegative quantity from the right part of (1.1). In this way, every result that stems from (1.1), such as the inequality (1.10) and its generalizations and refinements, can also be improved. Furthermore, we apply our improvements of inequalities (1.3), (1.6), and (1.7) to achieve refinements of some well-known trapezoid-Grüss and Ostrowki-Grüss type inequalities. Some refinements of inequality (1.8) as well as an additive reverse of the Schwarz inequality are also given.

## 2. A Refinement of Inequality (1.1)

Let be an inner product space over the real or complex number field . For our purpose, we need the following three lemmas.

Lemma 2.1 (see [2, page 599]).

For all , , one has

where is the Gramian of the vectors . The equality in (2.1) holds if and only if , or is linearly dependent, or is linearly dependent.

Lemma 2.2.

Let be an orthonormal system of vectors in , then, for any vectors , one has that

is linearly dependent, if and only if is linearly dependent.

Proof.

Let be linearly dependent, then clearly is also linearly dependent. Conversely, let be linearly dependent, then since is linearly independent, there exist and not all zero, such that

which for some can be rewritten as follows:

Hence, we get

Further, for , we have

Hence from (2.5), we get , Consequently, from (2.4) we obtain

and because at least one of is nonzero, it is derived that

is linearly dependent.

Lemma 2.3.

Let be an orthonormal system of vectors in and be any vectors in , then,

where

Proof.

Define the linear mapping

given by It is easy to verify that

which completes the proof.

Theorem 2.4.

Let be an orthonormal system of vectors in and let be linear independent, such that , then for all the following inequality holds:

where

and is given as in (1.2). The equality in (2.13) holds if and only if , or is linearly dependent.

Proof.

If we apply inequality (2.1) for by choosing , , , and and taking into consideration that for all , we have

then we get

Now, from the condition , the following is derived:

From the condition " is linear independent", it follows that . Now, if we set (2.17) and (2.18) in (2.16), we readily get the conclusion. Finally, according to Lemma 2.1 we have that the equality in (2.16) holds if and only if

or is linearly dependent, or is linearly dependent. Now, since , (2.19) can be rewritten as follows:

Moreover, according to Lemma 2.3, we have that

Combining (2.20) with (2.21), we conclude that (2.19) is equivalent to

From the conditions of this theorem, we clearly have that is linearly independent and since we obtain that is also linear independent. Finally, according to Lemma 2.2, we have that is linearly dependent if and only if is linearly dependent, which completes the proof.

Corollary 2.5.

Assuming that the conditions of Theorem 2.4 hold and then one has

The equality in (2.23) holds if and only if is linearly dependent.

Proof.

From the condition , we obtain that . Hence, (2.23) can be rewritten as follows:

Furthermore, from (2.13) and (2.14), we have

Consequently, we can apply the elementary inequality

for

to obtain

Combining (2.24) with (2.28) we get the desired result.

Letting , we easily derive that and , hence the equality in (2.23) holds. Conversely, it is clear that the equality in (2.23) holds if and only if the equalities in (2.24) and (2.28) hold. That is

Now, from (2.29) it follows that for some ,

Putting (2.31) and (2.32) in (2.30), we get after some algebraic calculations,

and since , we conclude that .

Therefore, there is in such that,

Putting (2.34) in (2.32), dividing the result by and finally subtracting this result from (2.31), we conclude that is linearly dependent.

Remark 2.6.

The main result in [11] is an identity, which by setting can be equivalently written in the following form:

It is hard to find a positive lower bound of the term . Therefore, it is difficult to derive a refinement of inequality (1.1) like (2.13) for from the above identity.

## 3. A Refinement of Dragomir's Inequality

The main result of this section is a refinement of Dragomir's inequality (1.10).

Theorem 3.1.

Let be a unit vector in If , , , and are real or complex numbers with , , and are vectors in satisfying the following conditions:

then for all nonproportional vectors such that the following inequality holds

where is as given in (2.14).

Proof.

We distinguish two cases.

Case 1.

Let either or . Without loss of generality, let us assume that Then from the condition , we have that . Thus, . So Consequently, (3.2) reduces to inequality (1.10).

Case 2.

Let , then we can apply inequality (2.23) by to obtain

Furthermore, from the conditions (3.1), we have the following (see [3]):

Finally, combining (3.3) with (3.4) leads to the asserted inequality (3.2).

Now, working similar as above we can show the following result, which we will use in the next section, to obtain refinements of some known integral inequalities.

Theorem 3.2.

Let be a unit vector in If are real or complex numbers with , and are vectors in satisfying the following condition:

then for all nonproportional vectors such that the following inequality holds:

Remark 3.3.

Based on inequality (2.23), one can improve, in a way similar to Theorem 3.1, all results related to Grüss inequality as in [1, 4–8] For example, in [1, pages 333-334], [7, page 2751], and [4, page 90], Dragomir obtained the following inequality by using Arcel's inequality:

which is used to derive the following refinement of (1.10)

Now, if we combine the inequalities (3.3), (3.7), and (3.4), we get the following improvement of (3.8):

## 4. A Refinement of Grüss Inequality and Applications

First we will use the results of Section 3 to obtain improvements of inequalities (1.3), (1.6), and (1.7).

Theorem 4.1.

Let be bounded on and let be not proportional and so that Then, provided that are nonconstant functions, the following inequalities hold:

where

and is the Chebyshev functional defined by (1.4).

Proof.

Applying inequality (2.23) on for as well as the inequalities (3.6) and (3.2) by choosing , , , , , we easily get the required inequalities.

Note that all known trapezoid-Grüss, midpoint-Grüss, and Ostrowski-Grüss type inequalities, which are proved by applying the inequalities (1.3), (1.6), and (1.7), can be improved by using the inequalities of Theorem 4.1. For example, in [12, page 39] we can see the following trapezoid-Grüss type inequality:

where is a twice differentiable mapping on such that , for all

In the following, we apply inequality (4.2) to derive a refinement of inequality (4.5).

Proposition 4.2.

Let , , and be as above, then

Proof.

Let be functions defined by , , , and , then it is easy to verify that

Finally, since equalities , hold, we can apply inequality (4.2), using the above relations to get the required inequality.

In [13, page 167] we can see the following Ostrowski-Grüss type inequality for all :

where is a differentiable function on such that , for all , and is defined by

We propose improvement of this result as follows.

Proposition 4.3.

Let be as above, then,

Proof.

If we apply inequality (4.2) by choosing

and take into account that, after some calculations, we obtain

and we easily derive the asserted inequality.

According to [13, page 168], the following inequality holds:

If we apply inequality (4.10) for , we get the following refinement of the previous inequality.

Corollary 4.4.

Let be as in Proposition 4.3, then,

It can now be observed that (4.19) can be written as follows:

or equivalently

which, by using inequality (2.26), leads to the following result.

Corollary 4.5.

Let be as above, then,

Now, we will use again inequality (4.2) to give another improvement of inequality (4.8).

Proposition 4.6.

Let be as in Proposition 4.3, then for all ,

where

Proof.

Let us choose , , and as given in (4.11) and (4.12). We distinguish the following two cases.

Case 1 ().

Then, by choosing

it can be easily verified that

So, we can apply inequality (4.2), by using (4.12), (4.14), (4.15), and (4.16) as well as

to obtain the desired inequality (4.22).

Case 2 ().

Then, by choosing

we easily derive the following:

Finally, application of inequality (4.2) using the above relations leads to the claimed result.

## 5. Refinements of Discrete Grüss Inequality

In this section, some refinements of the discrete version of Grüss inequality (1.8), as well as, an additive reverse of the Schwarz inequality are provided.

Theorem 5.1.

Let , be real numbers so that , for where and . Let be real numbers such that the vectors are not proportional and

Then,

Proof.

Define the vectors , , , , in the Euclidian inner product space . Then, we have

and . Hence, we can apply inequality (3.2) of Theorem 3.1 for the vectors , , , , and , as given above, to complete the proof.

Corollary 5.2.

Let be as is in Theorem 5.1, then for all ,

where

Proof.

If or or , then inequality (5.4) reduces to (1.8). If , , and , then by choosing the vectors , such that

it follows that these vectors satisfy the conditions of Theorem 5.1. Hence, we can apply Theorem 5.1 to obtain the desired result.

Now, if we apply Corollary 5.2 by choosing and , we immediately obtain the following result.

Corollary 5.3.

Let be real numbers so that , for all , where and . Then for all , one has the following inequality:

Corollary 5.4.

Let and , then

Proof.

If we apply inequality (5.7) by choosing , , , and , , we directly get the desired inequality.

Now, we will use inequality (5.7) to derive an additive reverse of the Schwarz inequality.

Applying times the inequality (5.7), namely for all pairs of integers with , adding the resulting inequalities, using the Lagrange identity, and finally dividing the resulting inequality by , then,

Now, if we solve inequality (5.9) with respect to and then replace by , we obtain the following reverse of the Schwarz inequality.

Proposition 5.5.

Let , , be real numbers so that , for all where and . then,

## References

- 1.
Dragomir SS: On Bessel and Grüss inequalities for orthonormal families in inner product spaces.

*Bulletin of the Australian Mathematical Society*2004, 69(2):327–340. 10.1017/S0004972700036066 - 2.
Mitrinović DS, Pečarić JE, Fink AM:

*Classical and New Inequalities in Analysis, Mathematics and Its Applications*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1993:xviii+740. - 3.
Dragomir SS: A generalization of Grüss inequality in inner product spaces and applications.

*Journal of Mathematical Analysis and Applications*1999, 237(1):74–82. 10.1006/jmaa.1999.6452 - 4.
Dragomir SS:

*Advances in Inequalities of the Schwarz, Grüss and Bessel Type in Inner Product Spaces*. Nova Science, Hauppauge, NY, USA; 2005:viii+249. - 5.
Dragomir SS: Some Grüss type inequalities in inner product spaces.

*Journal of Inequalities in Pure and Applied Mathematics*2003., 4(2, article 42): - 6.
Dragomir SS: Reverses of Schwarz, triangle and Bessel inequalities in inner product spaces.

*Journal of Inequalities in Pure and Applied Mathematics*2004., 5(3, article 76): - 7.
Dragomir SS: New inequalities of the Kantorovich type for bounded linear operators in Hilbert spaces.

*Linear Algebra and its Applications*2008, 428(11–12):2750–2760. 10.1016/j.laa.2007.12.025 - 8.
Elezović N, Marangunić L, Pečarić J: Unified treatment of complemented Schwarz and Grüss inequalities in inner product spaces.

*Mathematical Inequalities & Applications*2005, 8(2):223–231. - 9.
Ilišević D, Varošanec S: Grüss type inequalities in inner product modules.

*Proceedings of the American Mathematical Society*2005, 133(11):3271–3280. 10.1090/S0002-9939-05-07937-2 - 10.
Ujević N: A new generalization of Grüss inequality in inner product spaces.

*Mathematical Inequalities & Applications*2003, 6(4):617–623. - 11.
Ma J: An identity in real inner product spaces.

*Journal of Inequalities in Pure and Applied Mathematics*2007., 8(2, article 48): - 12.
Cerone p, Dragomir SS: Trapezoidal-type rules from an inequalities point of view.

*RGMIA Research Report Collection*1999., 2(6, article 8): - 13.
Matić M, Pečarić J, Ujević N: Improvement and further generalization of inequalities of Ostrowski-Grüss type.

*Computers & Mathematics with Applications*2000, 39(3–4):161–175. 10.1016/S0898-1221(99)00342-9

## Author information

## Rights and permissions

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## About this article

### Cite this article

Kechriniotis, A., Delibasis, K. On Generalizations of Grüss Inequality in Inner Product Spaces and Applications.
*J Inequal Appl* **2010, **167091 (2010). https://doi.org/10.1155/2010/167091

Received:

Revised:

Accepted:

Published:

### Keywords

- Real Number
- Linear Mapping
- Unit Vector
- Complex Number
- Differentiable Function