The grand Furuta inequality (GFI) is understood as follows: If positive operators A and B on a Hilbert space satisfy A B 0, A is invertible and t ∈ [0, 1] ,then A 1 .t + r (A r 2 (A . t 2 B p A . t 2 ) s A r 2 ) 1 .t + r ( p.t ) s + r holds for p, s 1andr t . In this note, we present a short proof of (GFI) which is done by the usual induction on s and the use of the Furuta inequality. Furthermore we propose another simultaneous extension of the Ando-Hiai and Furuta inequalities: If A B 0, A is invertible and t ∈ [0, 1] ,then A t 1 .t p.t B p A .r + t 1 .t + r ( p.t ) s + r (A t s B p ) holds for r t and p, s 1. Here α is the α -geometric mean and s for s ∈ [0, 1] is of the same form as α .
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机译：广义Furuta不等式（GFI）的理解如下：如果希尔伯特空间上的正算子A和B满足AB 0，则A是可逆的，并且t∈[0，1]，则A 1 .t + r（A r 2（ A。t 2 B p A。t 2）s A r 2）1 .t + r（pt）s + r对于p，s 1andr t成立。在本说明中，我们提供了（GFI）的简短证明，该证明是通过对s的通常归纳和使用Furuta不等式来完成的。此外，我们提出了Ando-Hiai和Furuta不等式的另一个同时扩展：如果AB 0，A是可逆的，并且t∈[0，1]，则A t 1 .t pt B p A .r + t 1 .t + r （pt）s + r（A ts B p）对于rt和p，s 1成立。这里α是α几何平均值，对于s∈[0，1]的s与α的形式相同。
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