Open Access

On Several Matrix Kantorovich-Type Inequalities

Journal of Inequalities and Applications20102010:571629

DOI: 10.1155/2010/571629

Received: 5 September 2009

Accepted: 1 February 2010

Published: 14 February 2010

Abstract

We present several matrix Kantorovich-type inequalities, which improve the results obtained in Liu and Neudecker (1996). Elementary methods suffice to prove the inequalities.

1. Introduction

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

A well-know matrix version of Kantorovich inequality asserts that (see[13])

(1.1)

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., [4]):

(1.2)

It is not difficult to see that if , then we can get ; thus, for

In paper [5], from (which is equivalent to (13) in [6]), Liu and Neudecker presented the following so-called Kantorovich-type inequality:

(1.3)

for and and the following inequality:

(1.4)

for and Furthermore, in the same way, they obtained three more general versions.

(1.5)
(1.6)
(1.7)

for and

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.

Lemma 2.1.

for and

Proof.

It is easy to see that if then thus, we have
(2.1)

for

In [7], Dragomir defines a transform for this transform, we have the following lemma.

Lemma 2.2.

Let then
(2.2)
thus
(2.3)

for and

Proof.

(2.4)

From Lemma 2.2, we can easily get the inequality (1.4).

Corollary 3.2.

for and

Proof.

From , we have
(2.5)
then
(2.6)

The proof is completed.

Theorem 2.4.

for and

Proof.

(2.7)
From Lemmas 2.1 and 2.2, we have
(2.8)

The proof of Theorem 2.4 is completed.

Remark 2.5.

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.

Theorem 2.6.

(2.9)
(2.10)
(2.11)

for and where

Proof.

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
(2.12)

Remark.

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.

Declarations

Acknowledgments

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.

Authors’ Affiliations

(1)
School of Mathematical Sciences, Xiamen University
(2)
Department of Mathematics, Jiujiang University
(3)
School of Mathematics and Computer Science, Guizhou Normal University

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Copyright

© Zhibing Liu et al. 2010

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