Estimates of Ando–Hiai-type inequalities on operator power means

ABSTRACT

We aim to give Ando–Hiai-type ratio inequalities for an operator power mean $P_t(\omega ;\mathbb{A})$ and its reverse ones which represent a relation between $P_t(\omega ;{\mathbb{A}}^r)$ and $P_t(\omega ;\mathbb{A})$ for r>0 under k-tuple of positive invertible operators $\mathbb{A}=(A_1,\dots ,A_k)$ and $t\in (0,1]$. As applications, we consider complementary inequalities related to the information monotonicity property via any unital positive linear map. In particular, we state some results for the Karcher means by $t\to 0$.

Keywords:

• Operator power means
• Ando–Hiai inequality
• positive linear map
• Karcher mean

2010 Mathematics Subject Classifications:

• Primary 47A64
• Secondary 47A63

Disclosure statement

No potential conflict of interest was reported by the authors.