- 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
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.
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.
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.
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.
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.
- 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/S0004972700036066MathSciNetView ArticleMATHGoogle Scholar
- 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.View ArticleGoogle Scholar
- 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.1999.6452MathSciNetView ArticleMATHGoogle Scholar
- Dragomir SS: Advances in Inequalities of the Schwarz, Grüss and Bessel Type in Inner Product Spaces. Nova Science, Hauppauge, NY, USA; 2005:viii+249.MATHGoogle Scholar
- Dragomir SS: Some Grüss type inequalities in inner product spaces. Journal of Inequalities in Pure and Applied Mathematics 2003., 4(2, article 42):Google Scholar
- 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):Google Scholar
- 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.2007.12.025MathSciNetView ArticleMATHGoogle Scholar
- 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.MathSciNetView ArticleMATHGoogle Scholar
- Ilišević D, Varošanec S: Grüss type inequalities in inner product modules. Proceedings of the American Mathematical Society 2005, 133(11):3271–3280. 10.1090/S0002-9939-05-07937-2MathSciNetView ArticleMATHGoogle Scholar
- Ujević N: A new generalization of Grüss inequality in inner product spaces. Mathematical Inequalities & Applications 2003, 6(4):617–623.MathSciNetView ArticleMATHGoogle Scholar
- Ma J: An identity in real inner product spaces. Journal of Inequalities in Pure and Applied Mathematics 2007., 8(2, article 48):Google Scholar
- Cerone p, Dragomir SS: Trapezoidal-type rules from an inequalities point of view. RGMIA Research Report Collection 1999., 2(6, article 8):Google Scholar
- 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)00342-9View ArticleMATHGoogle Scholar
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