- Research Article
- Open Access
On Several Matrix Kantorovich-Type Inequalities
© Zhibing Liu et al. 2010
- Received: 5 September 2009
- Accepted: 1 February 2010
- Published: 14 February 2010
We present several matrix Kantorovich-type inequalities, which improve the results obtained in Liu and Neudecker (1996). Elementary methods suffice to prove the inequalities.
- General Version
- Matrix Version
- Kantorovich Inequality
Let be an -by- matrix; the Moore-Penrose inverse of is defined as the unique -by- matrix satisfying all of the following four criteria (see, e.g., ):
In the next section, we shall present several similar matrix Kantorovich-type inequalities, which improve some results above.
We first introduce two lemmas.
In , Dragomir defines a transform for this transform, we have the following lemma.
From Lemma 2.2, we can easily get the inequality (1.4).
The proof is completed.
The proof of Theorem 2.4 is completed.
Furthermore, in similar way we got Theorem 2.4, and we obtain three more general versions, which also improve the inequalities (1.5), (1.6), (1.7), respectively.
In fact, they are equivalent by noting and For (2.9), pre- and postmultiplying by and respectively, we get the inequality (2.10); similarly, for (2.10), pre- and postmultiplying by respectively, we get the inequality (2.11). So, we only prove the inequality (2.9).
From the proof, it is easy to see that so, we conclude that the inequality (2.9) gives an improvement of the inequality (1.5), meanwhile, the inequalities (2.10) and (2.11) improve the inequalities (1.6) and (1.7), respectively.
The authors would like to thank the anonymous referees for their valuable comments which have been implemented in this revised version. This work is supported by Natural Science Foundation of China no.10961010, Natural Science Foundation of Jiangxi, China no 2007GZS1760, and scientific and technological project of Jiangxi education office, China no GJJ08432.
- Mond B, Pečarić JE: A matrix version of the Ky Fan generalization of the Kantorovich inequality. Linear and Multilinear Algebra 1994, 36(3):217–221. 10.1080/03081089408818291MathSciNetView ArticleMATHGoogle Scholar
- Wang SG, Shao J: Constrained Kantorovich inequalities and relative efficiency of least squares. Journal of Multivariate Analysis 1992, 42(2):284–298. 10.1016/0047-259X(92)90048-KMathSciNetView ArticleMATHGoogle Scholar
- Chen L, Zeng X-M: Rate of convergence of a new type kantorovich variant of bleimann-butzer-hahn operators. Journal of Inequalities and Applications 2009, 2009:-10.Google Scholar
- Golub GH, Van Loan CF: Matrix Computation. Johns Hopkins University, Baltimore, Md, USA; 1983.MATHGoogle Scholar
- Liu S, Neudecker H: Several matrix Kantorovich-type inequalities. Journal of Mathematical Analysis and Applications 1996, 197(1):23–26. 10.1006/jmaa.1996.0003MathSciNetView ArticleMATHGoogle Scholar
- Marshall AW, Olkin I: Matrix versions of the Cauchy and Kantorovich inequalities. Aequationes Mathematicae 1990, 40(1):89–93. 10.1007/BF02112284MathSciNetView ArticleMATHGoogle 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
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