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 ![](https://www.scirp.org/html/htmlimages\2-7401658x\34678b57-3955-4a10-9ca0-3508441c14c1.png)
If
, then there exists a unitary matrix
such as
![](https://www.scirp.org/html/htmlimages\2-7401658x\56cd89f8-4bc4-4ad8-934d-c05e90ec6061.png)
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
![](https://www.scirp.org/html/htmlimages\2-7401658x\b2476f52-e7cb-45ac-8159-de0472232dd3.png)
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
![](https://www.scirp.org/html/htmlimages\2-7401658x\78776611-2b48-414f-9376-b988802e4841.png)
Then
(4)
with equality if and only if ![](https://www.scirp.org/html/htmlimages\2-7401658x\ad9640d7-f9d0-4791-8b3f-54fec6d3bf30.png)
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
![](https://www.scirp.org/html/htmlimages\2-7401658x\6935e6bc-6ed2-4f07-99d4-ffe309c263be.png)
Lemma 2.2. ([13], p.50) Let
,
,
. If
and
are commute, then exists a unitary matrix
such that
![](https://www.scirp.org/html/htmlimages\2-7401658x\1e0b3234-3f95-46d5-aa86-59bf63908d16.png)
Lemma 2.3. ([14], p.35) Let
. Then
![](https://www.scirp.org/html/htmlimages\2-7401658x\f4f2c610-4a8d-457e-a46c-2f219a56f5d4.png)
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
![](https://www.scirp.org/html/htmlimages\2-7401658x\599b6b28-7d5b-4ef8-b0ad-dffa75aed6e3.png)
Hence,
![](https://www.scirp.org/html/htmlimages\2-7401658x\9b5be830-a68f-4228-9527-32408339ee3d.png)
By (2), we have
![](https://www.scirp.org/html/htmlimages\2-7401658x\0d2241fe-dd18-495e-9ddb-1cfba392ffef.png)
![](https://www.scirp.org/html/htmlimages\2-7401658x\0af0a810-6e4f-404c-9c1b-e386085e8134.png)
and
![](https://www.scirp.org/html/htmlimages\2-7401658x\fd768258-27df-4e69-afa0-f2b90f79175e.png)
Since
is a concave function, by lemma 2.3, we get
![](https://www.scirp.org/html/htmlimages\2-7401658x\3770193b-c11a-458a-9511-3f375431ed75.png)
(5)
(6)
![](https://www.scirp.org/html/htmlimages\2-7401658x\c6417dd3-cc80-4b42-bb6a-9035b8213e3b.png)
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
![](https://www.scirp.org/html/htmlimages\2-7401658x\89432396-2265-4c16-aeef-edf2eb559b55.png)
with equality if and only if ![](https://www.scirp.org/html/htmlimages\2-7401658x\ed705f58-0140-40b8-8adc-a95d8555c230.png)
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
![](https://www.scirp.org/html/htmlimages\2-7401658x\ac352a0f-9404-4743-b12e-281e63ab83db.png)
and
![](https://www.scirp.org/html/htmlimages\2-7401658x\d65ceae5-1ecd-476f-8aa3-3960bba30595.png)
So
![](https://www.scirp.org/html/htmlimages\2-7401658x\549e47c1-52ab-4b31-95cc-d3fc69e4cced.png)
![](https://www.scirp.org/html/htmlimages\2-7401658x\4eba5921-731c-4010-9ccc-fcd51d1cd4fb.png)
It is easy to see that (4) holds if and only if
(7)
Since
, by Lemma 2.1, we have
![](https://www.scirp.org/html/htmlimages\2-7401658x\f91be64b-044d-4751-8b55-74b82adcca3e.png)
Now we prove (7). Put
![](https://www.scirp.org/html/htmlimages\2-7401658x\b7ceb645-977a-4264-9181-754f1a68f507.png)
Then
![](https://www.scirp.org/html/htmlimages\2-7401658x\3883dbad-33bf-4013-b66e-e8707143f1c8.png)
Applying Minkowski inequality, we have
![](https://www.scirp.org/html/htmlimages\2-7401658x\e95f98b1-8e05-476d-a6bf-2288ed8be6ab.png)
Using the Lemma 2.3 to the right of the above inequlity, we obtain
![](https://www.scirp.org/html/htmlimages\2-7401658x\dcc8f645-c115-47c4-b1b7-74d14215b15d.png)
which implies that
![](https://www.scirp.org/html/htmlimages\2-7401658x\05b1f710-1e0a-44dd-93bb-8362946e1d61.png)
It follows that
![](https://www.scirp.org/html/htmlimages\2-7401658x\d58e0b17-97f0-4b48-8cd1-18cef561dd0a.png)
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
![](https://www.scirp.org/html/htmlimages\2-7401658x\fcf039f8-da5f-4b1c-95f6-fa8880178a6e.png)
Then
![](https://www.scirp.org/html/htmlimages\2-7401658x\1615b728-b29a-4c81-8666-5626668d3a59.png)
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).