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.
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)
Lemma 2.1 (see [2, page 599]).
is linearly dependent.
which completes the proof.
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.
3. A Refinement of Dragomir's Inequality
The main result of this section is a refinement of Dragomir's inequality (1.10).
We distinguish two cases.
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.
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).
In the following, we apply inequality (4.2) to derive a refinement of inequality (4.5).
We propose improvement of this result as follows.
and we easily derive the asserted 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).
to obtain the desired inequality (4.22).
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.
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, we will use inequality (5.7) to derive an additive reverse of the Schwarz inequality.
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