Improved Heinz inequality and its application

* Correspondence: limin-zou@163. com School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing 404100, People’s Republic of China Abstract We obtain an improved Heinz inequality for scalars and we use it to establish an inequality for the Hilbert-Schmidt norm of matrices, which is a refinement of a result due to Kittaneh. Mathematical Subject Classification 2010: 26D07; 26D15; 15A18.


Introduction
Let M n be the space of n × n complex matrices and ||·|| stand for any unitarily invariant norm on M n . So, ||UAV|| = ||A|| for all A M n and for all unitary matrices U, V M n . If A = [a ij ] M n , then is the Hilbert-Schmidt norm of matrix A. It is known that the Hilbert-Schmidt norm is unitarily invariant.
The classical Young's inequality for nonnegative real numbers says that if a, b ≥ 0 and 0 ≤ v ≤ 1, then (1:1) with equality if and only if a = b. Young's inequality for scalars is not only interesting in itself but also very useful. If v = 1 2 , by (1.1), we obtain the arithmetic-geometric mean inequality Kittaneh and Manasrah [1] obtained a refinement of Young's inequality as follows: where r 0 = min {v, 1 − v}. Let a, b ≥ 0 and 0 ≤ v ≤ 1. The Heinz means are defined as follows: It follows from the inequalities (1.1) and (1.2) that the Heinz means interpolate between the geometric mean and the arithmetic mean: (1:4) The second inequality of (1.4) is known as Heinz inequality for nonnegative real numbers.
As a direct consequence of the inequality (1.3), Kittaneh and Manasrah [1] obtained a refinement of the Heinz inequality as follows: where r 0 = min {v, 1 − v}. Bhatia and Davis [2] proved that if A, B, X M n such that A and B are positive semidefinite and if 0 ≤ v ≤ 1, then (1:6) This is a matrix version of the inequality (1.4). Kittaneh [3] proved that if A, B, X M n such that A and B are positive semidefinite and if 0 ≤ v ≤ 1, then where r 0 = min {v, 1 − v}. This is a refinement of the second inequality in (1.6).
In this article, we first present a refinement of the inequality (1.5). After that, we use it to establish a refinement of the inequality (1.7) for the Hilbert-Schmidt norm.

A refinement of the inequality (1.5)
In this section, we give a refinement of the inequality (1.5). To do this, we need the following lemma.
Lemma 2.1. [4,5] Let f(x) be a real valued convex function on an interval [a, b]. For any 3 4 ]. (2:1) Proof. It is known that as a function of v, H v (a, b) is convex and attains its mini- which is equivalent to That is, So, which is equivalent to That is, So, , and so which is equivalent to , and so which is equivalent to That is, This completes the proof. □ Now, we give a simple comparison between the upper bound for If v ∈ 1 4 , 3 4 , then So, the inequality (2.1) is a refinement of the inequality (1.5).

An application
In this section, we give a refinement of the inequality (1.7) for the Hilbert-Schmidt norm based on the inequality (2.1). Theorem 3.1. Let A, B, X M n such that A and B are positive semidefinite and suppose that Then where r 0 = min {v, 1 − v}.