In 1987, we established an operator inequality as follows; $A \ge B \ge 0 $ $\Longrightarrow (A^{\frac {r}{2}} A^p A^{\frac {r}{2}})^{\frac{1}{q}} \ge (A^{\frac {r}{2}} B^p A^{\frac {r}{2}})^{\frac{1}{q}}$ holds for (*) $ p \ge 0$, $q \ge 1$, $r \ge 0$ with $(1+r)q \ge p+r.$ It is an extension of Löwner-Heinz inequality. The purpose of this paper is to explain geometrical background of the domain by (*), and to give brief survey of recent results of its applications.