Generalizations of a Matrix Inequality

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


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][2][3][4][5][6][7][8]). The matrix form of the Brunn-Minkowski inequality (see [9,10]) asserts that if A and B are two positive definite matrices of order n and 0 1 λ < < , then ( ) ( ) with equality if and only if ( ) if A is a positive definite (positive semi-definite) matrix, and * A denotes the transpose of A .Let , 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 A , B be positive definite commuting matrix of order n with eigenvalues in the interval I .If f is a positive concave function on I and 0 with equality if and only if f is linear and ( ) ( ) ( ) ≥ .We can define the determinant differences function of A and B by ( ) The following theorem gives another generalization of (1).Theorem 1.2.Let A , B be positive definite commuting matrix of order n with eigenvalues in the interval I and 1 I ∈ .Let f be a positive function on I and a and b be two nonnegative real numbers such that , , , with equality if and only if .

Proofs of Theorems
To prove the theorems, we need the following lemmas: , , , and , , , .
, where ν is a constant.This is a special case of Maclaurin's inequality.Proof of Theorem 1.1.Since A and B are commuted, by lemma 2.2, there exists a unitary matrix , , , and , , , . .
( ) ( ) ( ) ( ) ( ) ( ) Now we consider the conditions of equality holds.Since f is a concave function, the equality of ( 5) holds if and only if f is linear.By the equality of Lemma 2.3, the equality of ( 6) holds if and only if ( ) ( ), . So the equality of (3) holds if and only if f 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 A , B be positive definite commuting matrix of order n with eigenvalues in the interval I . If f is a positive concave function on I and
with equality if and only if .
It is easy to see that (4) holds if and only if Since ( ) ( ) ( ) ( ) , by Lemma 2.1, we have Applying Minkowski inequality, we have ) Using the Lemma 2.3 to the right of the above inequlity, we obtain which is just the inequality (7).By the equality conditions of Minkowski inequality and Lemma 2. Taking for ( ) f t t = in Theorem 1.2, we obtain the following corollary.Corollary 2.5.[7] Let A , B be positive definite commuting matrix of order n and a and b be two nonnegative real numbers such that , .
where A denotes the determinant of A .Let n n ×  denote the set of n n × real symmetry matrices.Let n I denote n n × unit matrix.We use the notation ( )

3 ,
the equality (1.4) holds if and only if complete the proof of Theorem 1.2. 1 Corollary 2.4, we obtain the Fan Ky concave theorem.Proof of Theorem 1.2.As in the proof of Theorem 1.1, since A and B are commuted, by lemma 2.2, there exists a unitary matrix U such that