On the Lawson–Lim means and Karcher mean for positive invertible operators

This note aims to generalize the reverse weighted arithmetic–geometric mean inequality of n positive invertible operators due to Lawson and Lim. In addition, we make comparisons between the weighted Karcher mean and Lawson–Lim geometric mean for higher powers.


Introduction
Since the pioneering papers of Pusz and Woronowicz [18], Ando [1], and Kubo and Ando [11], an extensive theory of two-variable geometric mean has sprung up for positive operators: For two positive operators A and B, the operator geometric mean is defined by A B := A 1 2 (A - 1 2 BA -1 2 ) 1 2 A 1 2 for A > 0. But the n-variable case for n > 2 was a long standing problem and many authors studied the geometric mean of n-variable.
In 2004, Ando et al. [2] succeeded in the formulation of the geometric mean for n positive definite matrices, and they showed that it satisfies ten important properties. Definition 1.1 (Ando-Li-Mathias geometric mean [2]) Let A i (i = 1, 2, . . . , n) be positive definite matrices. Then the geometric mean G ALM (A 1 , A 2 , . . . , A n ) is defined by induction as follows: (ii) Assume that the geometric mean of any n -1-tuple of operators is defined. Let and let the sequences i , and it does not depend on i, then the geometric mean of n-matrices is defined as In [20], Yamazaki pointed out that the definition of the geometric mean by Ando, Li and Mathias can be extended to Hilbert space operators. Lawson and Lim [12] established a definition of the weighted version of the Ando-Li-Mathias geometric mean for n positive operators, we call it Lawson-Lim geometric mean G[n, t](A 1 , A 2 , . . . , A n ); see [12] for more details. In particular, G[n, 1 2 ] for t = 1 2 is the Ando-Li-Mathias geometric mean. Similarly, the weighted arithmetic mean is defined as follows: Note that the coefficient {t[n] i } depends on n and t only; see [6,19] for more details. Moreover, the weighted arithmetic-geometric-harmonic mean inequalities holds: Since then, another approach to generalizing the geometric mean to n-variables, depending on Riemannian trace metric, was the Karcher mean, which was studied by many researchers; see [13,14] and the references therein. Let A = A 1 , A 2 , . . . , A n ∈ P n and ω = (w 1 , w 2 , . . . , w n ) ∈ n . By computing appropriate derivatives as in [3], the ω-weighted Karcher mean of A, denoted by G K (ω; A), coincides with the unique positive definite solution of the Karcher equation In the case of two operators, A 1 , A 2 ∈ P, the Karcher mean coincides with the weighted

Weighted arithmetic and geometric means due to Lawson and Lim
In 2006, Yamazaki [20] obtained a converse of the arithmetic-geometric mean inequality of n-operators via Kantorovich constant. Soon after, Fujii el al. [6] also proved a stronger reverse inequality of the weighted arithmetic and geometric means due to Lawson and Lim of n-operators by the Kantorovich inequality.
In this section, we present the higher-power reverse inequalities of the weighted arithmetic and geometric means due to Lawson and Lim of n-operators, and several complements of the weighted geometric mean for n-variables have been established.
It is well known that A ≤ 1 is equivalent to A * A ≤ I. This fact plays an important role in the proofs of the theorems.
Then is a positive linear map such that (I) = I. The condition 0 < m ≤ A i ≤ M for i = 1, 2, . . . , n implies that m ≤ (A 1 ⊕ · · · ⊕ A n ) ≤ M.
We obtain the desired result.
Putting α = 2 in the inequality (2.8) implies the following. Next, we show the complements of the weighted geometric mean due to Lawson and Lim by virtue of the following lemma (see Corollary 2.12 in [10]) and we will generalize Lemma 2.8 and Lemma 2.9 in [19] in two following theorems.

Comparisons between the weighted Karcher mean and the Lawson-Lim geometric mean
In the final section, we make comparisons between the weighted Karcher mean and the Lawson-Lim geometric mean for higher powers. This is a fascinating work because the order relation can be preserved between higher-power operators by the Kantorovich constant. Remark 3.1 When p = 1, t = 1 2 , the inequality (3.1) reduces to the first inequality of Theorem 5.1 in [7].