Further refinements of reversed AM–GM operator inequalities

In this paper, we shall give further improvements of reversed AM–GM operator inequalities due to Yang et al. (Math. Slovaca 69:919–930, 2019) for matrices and positive linear map.


Introduction
Throughout this paper, let m, M be scalars and I be the identity operator. Other capital letters are used to denote the general elements of the C * algebra B(H) of all bounded linear operators acting on a Hilbert space (H, ·, · ). Also A ≥ 0 means that the operator A is positive. We say that for v ∈ [0, 1], respectively, denoted by A∇B and A B for brevity when v = 1 2 . The Kantorovich constant is defined by K(t, 2) = (t+1) 2 4t for t > 0. What's more, the relative operator entropy of A and B is defined as S(A|B) = A 1 2 log(A -1 2 BA -1 2 )A 1 2 . For A = (a ij ) ∈ M n , the Hilbert-Schmidt norm is defined by A 2 = n i,j=1 a 2 ij . As we all know that · 2 has the unitary invariance property: UAV 2 = A 2 for all A ∈ M n and unitary matrices U, V ∈ M n . The singular values of a matrix A is defined by s j (A), j = 1, 2, . . . , n, and arranged in a nonincreasing order.
In this paper, we shall give further improvements of (1.12) and (1.13) for positive linear maps. We also give some inequalities for Hilbert-Schmidt norms and determinants.

Main results
Firstly, we give some further refinements of the corresponding results in [12] for scalars and Hilbert-Schmidt norms. Before that, we state a lemma.
We have: .
Here, we completed the proof of Theorem 2.2 when 0 ≤ v ≤ 1 2 . Substituting a by b and v by 1v in (2.2) and (2.3), we can get (2.4) and (2.5) easily, so we omit the details.
The following corollary is a direct consequence of Theorem 2.2 by substituting a by a 2 and b by b 2 .
We have: Proof Since A and B are strictly positive-definite matrices, it follows by the spectral decomposition theorem that there exist unitary matrices U, V ∈ M n , such that h and the monotonicity of the function f (x) = log x (0 < x ≤ 1), we get where we completed the proof of (2.10). Using the same method, we can get (2.11)-(2.13) by (2.7)-(2.9), respectively, so we omit the details.
Next, we give further improvements of (1.12) and (1.13) for positive linear maps. But first, let us present the following lemmas that will be useful later.

Now by the functional calculus for the positive operator
Multiplying by A -1 2 both sides of inequality (2.23), we have Moreover, where the first inequality is by (2.19), the second is by (2.24), and the third is by (2.22). Therefore, where the first inequality is by (2.16) and the second is by (2.25). That is, By Lemma 2.6, we have When 0 ≤ p ≤ 2, then 0 ≤ p 2 ≤ 1, hence by (2.17), we have When p > 2, where the first inequality is by (2.16), the second is by (2.18), and the third is by (2.25). That is, which is equivalent to (2.20) by (2.15). Here we completed the proof of (2.20). We now prove (2.21) for 0 ≤ p ≤ 2. Indeed, where the first inequality is by (2.16), the second is by (2.14), and the third is by (2.25).
That is, where the first inequality is by (2.16), the second is by (2.18), the third is by (2.14), and the last inequality is by (2.25). That is, B -1 )) ≥ 0. In other words, our results can be regarded as further refinements of reversed AM-GM operator inequalities of [12].
By the same methods of Theorem 2.11, we can get further improvements of (1.12) and (1.13) for 1 4 ≤ v ≤ 1.

Corollary 2.13
Under the same conditions as in Theorem 2.11, we have: and and  In the end of this paper, we give some inequalities for determinants which were not mentioned in Yang's paper and its references. But first, we state a lemma.  The first inequality is obtained by (2.36), while the second by Lemma 2.14. Multiplying by (det A 2 n both sides of inequalities above, we can get the desired inequality (2.32) directly. Using the same technique above in (2.7)-(2.9), we can (2.33)-(2.35), respectively. To keep our paper concise, we omit the details.