\(A\geq B\geq 0\) ensures \((B^{\frac r2}A^pB^{\frac r2})^{\frac 1q}\geq(B^{\frac r2}B^pB^{\frac r2})^{\frac 1q}\) for \(r\geq 0\), \(p\geq 0\), \(q\geq 1\) with \((1+r)q\geq p+r\) and its recent applications (Q2708932)

As an ingenious extension of the famous Löwner-Heinz inequality: \(A\geq B\geq 0\) ensures \(A^{\alpha}\geq B^{\alpha}\) for \(\alpha\in [0,1]\), Furuta established the following result: If \(A\geq B\geq 0\), then for each \(r\geq 0\), NEWLINE\[NEWLINE\text{(i)}\quad (B^{\frac{r}{2}}A^{p} B^{\frac{r}{2}})^{\frac{1}{q}} \geq (B^{\frac{r}{2}}B^{p} B^{\frac{r}{2}})^{\frac{1}{q}} \text{ and (ii)}\quad (A^{\frac{r}{2}}A^{p} A^{\frac{r}{2}})^{\frac{1}{q}} \geq (A^{\frac{r}{2}}B^{p} A^{\frac{r}{2}})^{\frac{1}{q}}NEWLINE\]NEWLINE hold for \(p\geq 0\), \(q\geq 1\) with \((1+r)q\geq p+r\). This result is called Furuta inequality, and provides us with many applications to the following three fields: (A) operator inequalities, (B) norm inequalities and (C) operator equations.NEWLINENEWLINENEWLINEThe paper gives an elementary and simplified proof of the Furuta inequality, and also gives a brief survey of some of its applications. Especially, in this paper, a generalized Furuta inequality, applications to log majorization, an application of Aluthge transformation to \(p\)-hyponormal operators and generalized relative operator entropy are introduced.NEWLINENEWLINEFor the entire collection see [Zbl 0947.00027].