1. Introduction

Let M m,n be the space of m × n complex matrices and M n = M n,n . Let denote any unitarily invariant norm on M n . So, for all A∈M n and for all unitary matrices U,V∈M n . For A = (a ij )∈M n , the Hilbert-Schmidt norm of A is defined by

where tr is the usual trace functional and s 1 (A) ≥ s 2 (A) ≥ ... ≥ s n-1 (A) ≥ s n (A) are the singular values of A, that is, the eigenvalues of the positive semidefinite matrix , arranged in decreasing order and repeated according to multiplicity. The Hilbert-Schmidt norm is in the class of Schatten norms. For 1 ≤ p < ∝, the Schatten p-norm is defined as

For k = 1,...,n, the Ky Fan k-norm is defined as

It is known that these norms are unitarily invariant, and it is evident that each unitarily invariant norm is a symmetric guage function of singular values [1, p. 54-55].

Bhatia and Davis proved in [2] that if A,B,X∈M n such that A and B are positive semidefinite and if 0 ≤ r ≤ 1, then

(1.1)

Let A,B,X∈M n such that A and B are positive semidefinite. In [3], Zhan proved that

(1.2)

for any unitarily invariant norm and real numbers r,t satisfying 1 ≤ 2r ≤ 3,-2 < t ≤ 2. The case r = 1,t = 0 of this result is the well-known arithmetic-geometric mean inequality

Meanwhile, for r∈[0,1], Zhan pointed out that he can get another proof of the following well-known Heinz inequality

by the same method used in the proof of (1.2).

Let A,B,X∈M n such that A and B are positive semidefinite and suppose that

(1.3)

Then ψ is a convex function on [-1,1] and attains its minimum at v = 0 [4, p. 265].

In [5], for positive semidefinite n × n matrices, the inequality

(1.4)

was shown to hold for every unitarily invariant norm. Meanwhile, Bhatia and Kittaneh [5] asked the following.

Question

Let A,B∈M n be positive semidefinite. Is it true that

, ?

The case n = 2 is known to be true [5]. (See also, [1, p. 133], [6, p. 2189-2190], [7, p. 198].)

Obviously, if A,B∈M n are positive semidefinite and AB = BA, then we have , .

2. Some inequalities for unitarily invariant norms

In this section, we first utilize the convexity of the function

to obtain an inequality for unitarily invariant norms that leads to a refinement of the inequality (1.2). To do this, we need the following lemmas on convex functions.

Lemma 2.1

Let A,B,X∈M n such that A and B are positive semidefinite. Then, for each unitarily invariant norm, the function

is convex on [0,2] and attains its minimum at r = 1.

Proof

Replace v+1 by r in (1.3).□

Lemma 2.2

Let ψ be a real valued convex function on an interval [a,b] which contains (x1,x2). Then for x1 ≤ x ≤ x2, we have

(2.1)

Proof

Since ψ is a convex function on [a,b], for a ≤ x1 ≤ x ≤ x2 ≤ b, we have

This is equivalent to the inequality (2.1).□

Theorem 2.1

Let A,B,X∈M n such that A and B are positive semidefinite. If 1 ≤ 2r ≤3 and -2 <t ≤ 2, then

(2.2)

where r0 = min{r,2-r}.

Proof

If , then by Lemma 2.1 and Lemma 2.2, we have

That is

(2.3)

It follows from (1.2) and (2.3) that

If , then by Lemma 2.1 and Lemma 2.2, we have

That is

(2.4)

It follows from (1.2) and (2.4) that

It is equivalent to the following inequality

This completes the proof.□

Now, we give a simple comparison between the upper bound in (1.2) and the upper bound in (2.2).

Therefore, Theorem 2.1 is a refinement of the inequality (1.2).

Let A,B,X∈M n such that A and B are positive semidefinite. Then, for each unitarily invariant norm, the function

is a continuous convex function on [0,1] and attains its minimum at . See [4, p. 265]. Then, by the same method above, we have the following result.

Theorem 2.2.[8]

Let A,B,X∈M n such that A and B are positive semidefinite. If 0 ≤ v ≤ 1, then

where r0 = min{v,1-v}. This is a refinement of the second inequality in (1.1).

Next, we will obtain an improvement of the inequality (1.4) for the Hilbert-Schmidt norm. To do this, we need the following lemma.

Lemma 2.3.[9]

Let A,B,X∈M n such that A and B are positive semidefinite. If 0 ≤ v ≤ 1, then

Theorem 2.3

Let A,B,X∈M n such that A and B are positive semidefinite. If 0 ≤ v ≤ 1, then

Proof

Let

So,

By Lemma 2.3, we have

That is,

Hence,

This completes the proof.□

Let A,B,X∈M n such that A and B are positive semidefinite, for Hilbert-Schmidt norm, the following equality holds:

Taking in Theorem 2.3, and then we have the following result.

Theorem 2.4.[10]

Let A,B,X∈M n such that A and B are positive semidefinite. Then

Bhatia and Kittaneh proved in [5] that if A,B∈M n are positive semidefinite, then

(2.5)

Now, we give an improvement of the inequality (1.4) for the Hilbert-Schmidt norm.

Theorem 2.5

Let A,B∈M n be positive semidefinite. Then

Proof

Let

Then, by Theorem 2.4, we have

(2.6)

It follows form (2.5) and (2.6) that

That is,

This completes the proof.□