Abstract
Suppose that A, B ∈ 𝔹(𝓗) are positive invertible operators. In this paper, we show that
$$\begin{array}{}
\displaystyle
A \# B \leq \frac{1}{1-2\mu}A^\frac{1}{2}F_\mu(A^\frac{-1}{2}BA^\frac{-1}{2})A^\frac{1}{2}\\
\displaystyle\qquad~\,\leq\frac{1}{2}\bigg[ A \# B +H_\mu (A,B)\bigg]\\
\displaystyle\qquad~\,\leq\frac{1}{2}\bigg[ \frac{1}{1-2\mu}A^\frac{1}{2}F_\mu(A^\frac{-1}{2}BA^\frac{-1}{2})A^\frac{1}{2}+H_\mu (A,B)\bigg]\\
\displaystyle\qquad~\,\leq \dots \leq \frac{1}{2^n}A \# B + \frac{2^n-1}{2^n}H_\mu (A,B)\\
\displaystyle\qquad~\,\leq \frac{1}{2^n(1-2\mu)}A^\frac{1}{2}F_\mu(A^\frac{-1}{2}BA^\frac{-1}{2})A^\frac{1}{2}+\frac{2^n-1}{2^n}H_\mu (A,B)\\
\displaystyle\qquad~\,\leq \frac{1}{2^{n+1}} A \# B +\frac{2^{n+1}-1}{2^{n+1}}H_\mu (A,B)\\
\displaystyle\qquad~\,\leq \dots \leq H_\mu (A,B)
\end{array}$$
for each
$\begin{array}{}
\displaystyle
\mu \in [0,1]\smallsetminus\{\frac{1}{2}\},
\end{array}$ where Hμ (A, B) and A#B are the Heinz mean and the geometric mean for operators A, B, respectively, and
$\begin{array}{}
\displaystyle
F_{\mu}\in C({\rm sp}(A^\frac{-1}{2}BA^\frac{-1}{2}))
\end{array}$ is a certain parameterized class of functions. As an application, we present several inequalities for unitarily invariant norms.
Harnack type inequalities for operators in logarithmic submajorisation
