Let \(A,B\) be bounded linear operators acting on a Hilbert space, \(A \geq B >0\), \(t \in[0,1]\), \(r \geq t\) and \(p, s \geq 1\). The well-known Grand Furuta (GF) inequality says that \[ A^{-r+t}\sharp_{\frac{1-t+r}{(p-t)s+r}} (A^{t}\natural_s B^p) \leq A \] (cf.\ \textit{T. Furuta} [Linear Algebra Appl. 219, 139--155 (1995; Zbl 0822.15008)]). A mean theoretic expression of it induces a satellite of the Grand Furuta (SGF) inequality as \[ A^{-r+t}\sharp_{\frac{1-t+r}{(p-t)s+r}} (A^{t}\natural_s B^p) \leq B \] (cf.\ \textit{M. Fujii} and \textit{E. Kamei} [Sci. Math. Jpn. 56, No. 3, 501--504 (2002; Zbl 1024.47004)]). Here, for \(\alpha \in [0,1]\), \(A\sharp_{\alpha} B\) is the \(\alpha\)-geometric mean defined by \(A^{\frac{1}{2}}(A^{\frac{-1}{2}}BA^{\frac{-1}{2}})^{\alpha}A^{\frac{1}{2}}\) and \(\natural_s\) is a formal extension of \(\sharp_\alpha\). In the paper under review, the authors show that \((SGF; 0\leq t \leq 1)\) has the Löwner-Heinz property, i.e., \((SGF; t = 1)\) implies that \((SGF; t)\) for every \(t \in [0, 1]\). They show that a recent further extension of (GFI) by \textit{T. Furuta} [J. Math. Inequal. 2, No. 4, 465--472 (2008; Zbl 1168.47016)] also has the Löwner-Heinz property.