On a family of operator means involving the power difference means

Abstract

A new family of operator means is introduced. It interpolates the arithmetic, geometric, harmonic and logarithmic means. Moreover it includes some special operator means, for example, the power difference, Stolarsky and identric means, continuously. Then order relations among them are obtained.

Spectral inequalities for Kubo-Ando operator means

“…For A, B ≥ 0 and X ∈ M n , we have A 1/2 B X A 1/2 ≥ 0, so that (11) λ 1 (AB X ) ≤ AB X and λ n (AB X ) ≥ s n (AB X ).…”

“…For r, s ∈ [−1, 1], the extension σ r,s of the power difference mean [11] is the operator mean with representing function F r,s (t), which is symmetric and σ ⊥ r,s = σ −r,−s . This family interpolates well-known operator means.…”

Further mean inequalities via some integral inequalities

“…As usual, f (A) is defined by the techniques of functional calculus. For further details about operator monotone functions we can consult [2,11,29,30] and the related references cited therein.…”

“…Utilizing the Ostrowski's inequality (29), we obtain another estimation of L(a, b) − a λ b as recited in what follows.…”

An integral representation of operator means via the power means and an application to the Ando-Hiai inequality

“…F p,q (x) is a representing function of the extension of the power difference mean. In [8], F p,q is monotone increasing on each parameter p, q ∈ [−1, 1] (the cases p = 0 and q = 0 are defined by taking limits). Then we have 1 + x 2 = F 1,1 (x)…”