AMApplied Mathematics2152-7385Scientific Research Publishing10.4236/am.2014.53034AM-42640ArticlesPhysics&Mathematics Generalizations of a Matrix Inequality ingzhiZhao1JunYuan2*YunfengCai3College of Science, Nanjing University of Posts and Telecommunications, NanjingSchool of Mathematics and Information Technology, Nanjing Xiaozhuang University, NanjingCollege of Teacher Education, Nanjing Xiaozhuang University, Nanjing* E-mail:yuanjun_math@126.com(JY);070220140503337341© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

In this paper, some new generalizations of the matrix form of the Brunn-Minkowski inequality are presented.

Brunn-Minkowski Inequality; Positive Definite Matrix; Determinant Differences
1. Introduction

The well-known Brunn-Minkowski inequality is one of the most important inequalities in geometry. There are many other interesting results related to the Brunn-Minkowski inequality (see [1-8]). The matrix form of the Brunn-Minkowski inequality (see [9,10]) asserts that if and are two positive definite matrices of order and, then

with equality if and only if, where denotes the determinant of.

Let denote the set of real symmetry matrices. Let denote unit matrix. We use the notation if is a positive definite (positive semi-definite) matrix, and denotes the transpose of. Let, then if and only if

If, then there exists a unitary matrix such as where is a diagonal matrix, and are the eigenvalues of, each appearing as its multiplicity. Assume now that is well defined. Then may be defined by (see e.g. [11, p. 71] or [12, p. 90])

In this paper, some new generalizations of the matrix form of the Brunn-Minkowski inequality are presented. One of our main results is the following theorem.

Theorem 1.1. Let, be positive definite commuting matrix of order with eigenvalues in the interval. If is a positive concave function on and, then

with equality if and only if is linear and.

Let, if. We can define the determinant differences function of and by The following theorem gives another generalization of (1).

Theorem 1.2. Let, be positive definite commuting matrix of order with eigenvalues in the interval and. Let be a positive function on and a and b be two nonnegative real numbers such that Then

with equality if and only if

Remark 1. Let in Theorem 1.1 or let and in Theorem 1.2. We can both obtain (1). Hence Theorem 1.1 and Theorem 1.2 are generalizations of (1).

2. Proofs of Theorems

To prove the theorems, we need the following lemmas:

Lemma 2.1. (, p.472) Let,. Then Lemma 2.2. (, p.50) Let, ,. If and are commute, then exists a unitary matrix such that Lemma 2.3. (, p.35) Let. Then with equality if and only if, where is a constant.

This is a special case of Maclaurin’s inequality.

Proof of Theorem 1.1.

Since and are commuted, by lemma 2.2, there exists a unitary matrix such that Hence, By (2), we have  and Since is a concave function, by lemma 2.3, we get  Now we consider the conditions of equality holds. Since is a concave function, the equality of (5) holds if and only if is linear. By the equality of Lemma 2.3, the equality of (6) holds if and only if which means. So the equality of (3) holds if and only if is linear and. This completes the proof of the Theorem 1.1.

Applying the arithmetic-geometric mean inequality to the right side of (3), we get the following corollary.

Corollary 2.4. Let, be positive definite commuting matrix of order with eigenvalues in the interval. If is a positive concave function on and, then with equality if and only if

Taking for in Corollary 2.4, we obtain the Fan Ky concave theorem.

Proof of Theorem 1.2.

As in the proof of Theorem 1.1, since and are commuted, by lemma 2.2, there exists a unitary matrix such that and So  It is easy to see that (4) holds if and only if

Since, by Lemma 2.1, we have Now we prove (7). Put Then Applying Minkowski inequality, we have Using the Lemma 2.3 to the right of the above inequlity, we obtain which implies that It follows that which is just the inequality (7).

By the equality conditions of Minkowski inequality and Lemma 2.3, the equality (1.4) holds if and only if, which means. Thus we complete the proof of Theorem 1.2.

Taking for in Theorem 1.2, we obtain the following corollary.

Corollary 2.5.  Let, be positive definite commuting matrix of order and a and b be two nonnegative real numbers such that Then with equality if and only if.

Acknowledgements

The authors are most grateful to the referee for his valuable suggestions. And the authors would like to acknowledge the support from the National Natural Science Foundation of China (11101216,11161024), Qing Lan Project and the Nanjing Xiaozhuang University (2010KYQN24, 2010KYYB13).

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