Improved Heinz inequality and its application

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}.