Abstract:

Furuta showed that if A ≥ B ≥0, then for each r≥0, f(p)=(A^r/2B^pA^r/2)^t+r/p+r
is decreasing for p≥t≥0. Using this result, the following inequality (C^r/2AB²A)^δC^r/2)^p-1+r/4δ+r≤C^p-1+r is obtained for 0<p≤1, r≥1, 1/4≤δ≤1 and three positive operators A, B, C satisfy (A^1/2 B A^1/2)^p/2 ≤ A^p, (B^1/2 AB^1/2)^p/2 ≥ B^p, (C^1/2 AC^1/2)^p/2 ≤C^p, (A^1/2 CA^1/2)^p/2≥A^p.

Key words: Positive operator, Furuta inequality, operator inequality