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).

To prove the theorems, we need the following lemmas:

Lemma 2.1. ([13], p.472) Let,. Then

Lemma 2.2. ([13], p.50) Let, ,. If and are commute, then exists a unitary matrix such that

Lemma 2.3. ([14], 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. [7] 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.

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).