^{1}

^{2}

^{*}

^{3}

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

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

with equality if and only if

Let

If

where

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

with equality if and only if

Let

The following theorem gives another generalization of (1).

Theorem 1.2. Let

Then

with equality if and only if

Remark 1. Let

To prove the theorems, we need the following lemmas:

Lemma 2.1. ([

Lemma 2.2. ([

Lemma 2.3. ([

with equality if and only if

This is a special case of Maclaurin’s inequality.

Proof of Theorem 1.1.

Since

Hence,

By (2), we have

and

Since

Now we consider the conditions of equality holds. Since

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

Corollary 2.4. Let

with equality if and only if

Taking for

Proof of Theorem 1.2.

As in the proof of Theorem 1.1, since

and

So

It is easy to see that (4) holds if and only if

Since

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

Taking for

Corollary 2.5. [

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).