We present several Ando-Hiai type inequalities for n-variable operator means for positive invertible operators. Ando-Hiai's inequalities given here are not only of the original type but also of the complementary type and of the reverse type involving the generalized Kantorovich constant.
Multivariate meansThroughout the paper, H is a general Hilbert space assumed to be infinite-dimensional unless otherwise stated, B(H) is the algebra of all bounded linear operators on H, B(H) + the set of positive operators in B(H), and P = P(H) the set of positive invertible operators in B(H). For X ∈ P, X ∞ is the operator norm of X and λ min (X) is the minimum of the spectrum of X, that is, λ min (X) = X −1 −1 ∞ . Moreover, I denotes the identity operator, and SOT means the strong operator topology on B(H). The Thompson metric d T on P is defined bywhere M(A/B) := inf{α > 0 : A ≤ αB}. It is known [26] that (P, d T ) is a complete metric space and both topologies on P induced by d T and · ∞ coincide.
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