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
(1)
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])
(2)
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
(3)
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
(4)
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. ([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
(5)
(6)
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
(7)
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
.
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).