Comprehensive survey on an order preserving operator inequality (Q1943492)

The paper contains a brief survey on recent results on a certain operator inequality and its applications, namely, \(A \geq B \geq 0 \Longrightarrow (A^{\frac {r}{2}} A^p A^{\frac {r}{2}})^{\frac{1}{q}} \geq (A^{\frac {r}{2}} B^p A^{\frac {r}{2}})^{\frac{1}{q}}\) holds for \(p\geq 0\), \(q\geq 1\), \(r\geq 0\) with \((1+r)q \geq p+r \). This is an extension of the Löwner-Heinz inequality. The author gives a geometrical interpretation of the above inequality.