A satellite of the grand Furuta inequality and its application

The grand Furuta inequality has the following satellite (SGF;$t\in 0\text{,}1]$), given as a mean theoretic expression:$A?B>0\text{,}t\in 0\text{,}1]?A^{-r+t}{\#}_{\frac{1-t+r}{(p-t)s+r}}(A^t{?}_s{B}^p)?B\text{for}r?t\text{;}p\text{,}s?1\text{,}$where ${\#}_{\alpha }$ is the $\alpha$-geometric mean and ${?}_s$ ($s?0\text{,}1]$) is a formal extension of ${\#}_{\alpha }$. It is shown that (SGF; $t\in 0\text{,}1]$) has the Löwner–Heinz property, i.e. (SGF; $t=1$) implies (SGF;t) for every $t\in 0\text{,}1]$. Furthermore, we show that a recent further extension of (GFI) by Furuta himself has also the Löwner–Heinz property.