A satellite of the grand Furuta inequality and its application

The grand Furuta inequality has the following satellite (SGF;t∈[0,1]), given as a mean theoretic expression:A⩾B>0,t∈[0,1]⇒A-r+t#1-t+r(p-t)s+r(At♮sBp)⩽Bforr⩾t;p,s⩾1,where #α is the α-geometric mean and ♮s (s∉[0,1]) is a formal extension of #α. It is shown that (SGF; t∈[0,1]) has the Löwner-Heinz property, i.e. (SGF; t=1) implies (SGF;t) for every t∈[0,1]. Furthermore, we show that a recent further extension of (GFI) by Furuta himself has also the Löwner-Heinz property.