- Research Article
- Open Access
On Generalizations of Grüss Inequality in Inner Product Spaces and Applications
© Aristides I. Kechriniotis and Konstantinos K. Delibasis. 2010
- Received: 23 December 2009
- Accepted: 15 June 2010
- Published: 6 July 2010
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.
- Real Number
- Linear Mapping
- Unit Vector
- Complex Number
- Differentiable Function
where are real numbers so that , for all , and , .
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.
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]).
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.
is linearly dependent, if and only if is linearly dependent.
is linearly dependent.
which completes the proof.
and is given as in (1.2). The equality in (2.13) holds if and only if , or is linearly dependent.
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.
The equality in (2.23) holds if and only if is linearly dependent.
Combining (2.24) with (2.28) we get the desired result.
and since , we conclude 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.
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.
The main result of this section is a refinement of Dragomir's inequality (1.10).
where is as given in (2.14).
We distinguish two cases.
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).
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.
First we will use the results of Section 3 to obtain improvements of inequalities (1.3), (1.6), and (1.7).
and is the Chebyshev functional defined by (1.4).
Applying inequality (2.23) on for as well as the inequalities (3.6) and (3.2) by choosing , , , , , we easily get the required inequalities.
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).
Finally, since equalities , hold, we can apply inequality (4.2), using the above relations to get the required inequality.
We propose improvement of this result as follows.
and we easily derive the asserted inequality.
If we apply inequality (4.10) for , we get the following refinement of the previous inequality.
which, by using inequality (2.26), leads to the following result.
Now, we will use again inequality (4.2) to give another improvement of inequality (4.8).
Let us choose , , and as given in (4.11) and (4.12). We distinguish the following two cases.
Case 1 ( ).
to obtain the desired inequality (4.22).
Case 2 ( ).
Finally, application of inequality (4.2) using the above relations leads to the claimed result.
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.
and . Hence, we can apply inequality (3.2) of Theorem 3.1 for the vectors , , , , and , as given above, to complete the proof.
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.
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.
Now, if we solve inequality (5.9) with respect to and then replace by , we obtain the following reverse of the Schwarz inequality.
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