Unitarily invariant norm inequalities for some means

We introduce some symmetric homogeneous means, and then show unitarily invariant norm inequalities for them, applying the method established by Hiai and Kosaki. Our new inequalities give the tighter bounds of the logarithmic mean than the inequalities given by Hiai and Kosaki. Some properties and norm continuities in parameter for our means are also discussed.


Introduction
In the previous paper, we derived the tight bounds for logarithmic mean in the case of Frobenius norm, inspired by the work of Zou in [1].  Our bounds for the logarithmic mean have improved the famous results by Hiai and Kosaki [3,4] in the special case, since Frobenius norm is one of unitarily invariant norms.
In this paper, we give the tighter bounds for the logarithmic mean than those by Hiai and Kosaki [3,4] for every unitarily invariant norm. That is, we give the generalized results of Theorem 1.1 for the unitarily invariant norm. For this purpose, we firstly introduce two quantities. Definition 1.3 For α ∈ R and x, y > 0, we set We note that we have the following bounds of logarithmic mean with the above two means (See Appendix in the paper [2]): where the logarithmic mean is defined by x, (x = y). (1) We here define a few symmetric homogeneous means using P α (x, y) and Q α (x, y) in the following way.
Definition 1.4 (i) For |α| ≤ 1 and x = y, we define, (ii) For α ∈ R and x = y, we define, (iii) For |α| ≤ 2 and x = y, we define, (iv) For |α| ≤ 1 and x = y, we define, We have the following relations for the above means: and H α (x, y) = G 2α (x, y). In addition, the above means are written as the following geometric bridges: y) and B α (x, y) are called Stolarsky mean and binomial mean, respectively.
In the previous paper [2], as tight bounds of logarithmic mean, the scalar inequalities were shown which equivalently implied Frobenius norm inequalities (Theorem 1.1). See Theorem 2.2 and Theorem 3.2 in [2] for details. In this paper, we give unitarily invariant norm inequalities which are general results including Frobenius norm inequalities as a special case.

Unitarily invariant norm inequalities
To obtain unitarily invariant norm inequalities, we apply the method established by Hiai and Kosaki [4,6,7,8].    The functions A α (x, y), L α (x, y), G α (x, y), H α (x, y) defined in Definition 1.4 are symmetric homogeneous means. We give powerful theorem to obtain unitarily invariant norm inequalities. In the references [4,6,7,8], another equivalent conditions were given. However here we give minimum conditions to obtain our results in this paper. Throughout this paper, we use the symbol B(H) as the set of all bounded linear operators on a separable Hilbert space H. We also use the notation K ≥ 0 if K ∈ B(H) satisfies Kx, x ≥ 0 for all x ∈ H (then K is called a positive operator).
Thanks to Theorem 2.2, our task to obtain unitarily invariant norm inequalities in this paper is to show the relation M N which is stronger than the usual scalar inequalities M ≤ N .
We firstly give monotonicity of three means H α (x, y), G α (x, y) and A α (x, y) for the parameter we consider the case α ≥ 0. Then we have the following proposition. Proof: This is a positive definite function for the case α < β, so that we have H β H α .
It may be notable that (iii) of the above proposition can be proven by the similar argument in Theorem 2.1 of the paper [4].
Next we give the relation among four means H α (x, y), G α (x, y), L α (x, y), and A α (x, y).

Norm continuity in parameter
In this section, we consider the norm continuity argument with respect to the parameter on our introduced means. Since we have the relation H α (x, y) = G 2α (x, y), we firstly consider the norm continuity in parameter on G α (S, T ).
We secondly consider the norm continuity in parameter on A α (S, T ).
From the inequality (4), we have Thus the right hand side of the inequality (5) is bounded from the above: Thus we have the inequality (3).
We also have the following proposition.