- Research Article
- Open Access
On Several Matrix Kantorovich-Type Inequalities
Journal of Inequalities and Applications volume 2010, Article number: 571629 (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.
Let be a positive (semi-)definite Hermite matrix with eigenvalues contained in the interval where Let be matrix, and let denotes the column space of
for and where denotes the conjugate transpose of the matrix .
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., ):
It is not difficult to see that if , then we can get ; thus, for
for and and the following inequality:
for and Furthermore, in the same way, they obtained three more general versions.
In the next section, we shall present several similar matrix Kantorovich-type inequalities, which improve some results above.
2. New Matrix Kantorovich-Type Inequalities
We first introduce two lemmas.
It is easy to see that if then thus, we have
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).
From , we have
The proof is completed.
From Lemmas 2.1 and 2.2, we have
The proof of Theorem 2.4 is completed.
It is not difficult to see that if then we conclude that Theorem 2.4 gives an improvement of the Kantorovich inequality (1.3).
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.
for and where
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).
Similarly, with Lemma 2.2, we have
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.
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. 10.1080/03081089408818291
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-K
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.
Golub GH, Van Loan CF: Matrix Computation. Johns Hopkins University, Baltimore, Md, USA; 1983.
Liu S, Neudecker H: Several matrix Kantorovich-type inequalities. Journal of Mathematical Analysis and Applications 1996, 197(1):23–26. 10.1006/jmaa.1996.0003
Marshall AW, Olkin I: Matrix versions of the Cauchy and Kantorovich inequalities. Aequationes Mathematicae 1990, 40(1):89–93. 10.1007/BF02112284
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
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.
About this article
Cite this article
Liu, Z., Lu, L. & Wang, K. On Several Matrix Kantorovich-Type Inequalities. J Inequal Appl 2010, 571629 (2010). https://doi.org/10.1155/2010/571629
- General Version
- Matrix Version
- Kantorovich Inequality