Refinements of the heinz inequalities for operators and matrices

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.