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Let A ≥ B ≥ 0 with A > 0 and let t ∈ [0, 1] and q ≥ 0. As a generalization of a result due to Furuta, it is shown that the operator function Gp,q,t(A, B, r, s)=A −r/2{Ar/2(A−t/2BpA−t/2)sAr/2}(q−t+r)/[(p−t)s+r]A−r/2 is decreasing for r ≥ t and s ≥ 1 if p ≥ max{q, t}. Moreover, if 1 ≥ p > t and q ≥ t, then Gp,q,t(A, B, r, s) is decreasing for r ≥ 0 and s ≥ q−t p−t . The latter result is an extension of an earlier result of Furuta.

A decreasing operator function associated with the Furuta inequality