Abstract
The Furuta inequality is known as an exquisite extension of the Lowner- Heinz inequality. The grand Furuta inequality is its further extension including Ando- Hiai inequality on log-majorization. We introduce two mean theoretic operator functions defined for positive invertible operators X ≤ A, for a given positive invertible operator A and given t ∈ (0, 1) : Φ(X)(= Φ(u, p; X)) = Auu+1 u+p X p (≤ X) for u ≥ 0 ,p ≥ 1 and Ψ(X)(= Ψ(q, s; X)) = (A tsX q ) 1 (1−s)t+sq (≤ X) for q ≥ 1 ,s ≥ 1. We have the grand Furuta inequality by using their composition and also have its further extension presented recently by Furuta, by using a successive composition of them.
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