Inequalities for the Heinz Mean of Sector Matrices
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Inequalities for the Heinz Mean of Sector Matrices Yanling Mao1 · Yaping Mao2 Received: 7 November 2019 / Accepted: 21 January 2020 © Iranian Mathematical Society 2020
Abstract In this paper, we extend the famous Heinz mean inequality to sector matrix case. We obtain four inequalities including a matrix inequality, a singular value inequality, a determinant inequality and an unitarily invariant norm inequality. Keywords Heinz mean · Sector matrix · Singular values · Inequality Mathematics Subject Classification 15A45 · 15A60
1 Introduction For 0 ≤ ν ≤ 1, the Heinz mean of two positive numbers a, b is defined as Hν (a, b) =
a ν b1−ν + a 1−ν bν . 2
For each pair (a, b) of positive numbers, the function Hν (a, b) of ν is symmetric about the point ν = 1/2 and attains its minimum value there. It is well known that √ a+b . ab ≤ Hν (a, b) ≤ 2
(1.1)
Communicated by Mohammad B. Asadi.
B
Yanling Mao [email protected] Yaping Mao [email protected]
1
School of Mathematics and Statistics, Qinghai Nationalities University, Xining 810007, Qinghai, China
2
School of Mathematics and Information Science, Qinghai Normal University, Xining 810008, Qinghai, China
123
Bulletin of the Iranian Mathematical Society
We are interested in studying the analogues of (1.1) for matrices. Recent studies about the Heinz mean can be found in [4,7,13]. Let Mn be the set of n × n complex matrices. For any A ∈ Mn , A∗ stands for the conjugate transpose of A. Every A ∈ Mn can be written as A = A + iA, where A = (A + A∗ )/2, A = (A − A∗ )/2i. Define a sector Sα on the complex plane Sα = {z ∈ C| z > 0, |z| ≤ (z) tan α}, where α ∈ [0, π/2) is fixed. The numerical range of A ∈ Mn is defined as the set on the complex plane [5] W (A) = {x ∗ Ax| x ∈ Cn , x = 1}. If W (A) ⊂ Sα , then A is called a sector matrix [9]. Clearly, if W (A) ⊂ Sα , then A is positive definite. Recent studies of sector matrices can be found in [6,8–11,14,15, 17,18]. The geometric mean has been defined for positive definite matrices. If A, B ∈ Mn are positive definite, we define their weighted geometric mean as Aν B = A1/2 (A−1/2 B A−1/2 )ν A1/2 , 0 ≤ ν ≤ 1. When ν = 1/2, we write AB instead of A1/2 B. Recently, Raissouli, Moslehian and Furuichi [12] defined the weighted geometric mean for A, B ∈ Mn with A and B positive definite (such A, B are called accretive in the literature): sin νπ Aν B = π
∞
s ν−1 (A−1 + s B −1 )−1 ds,
(1.2)
0
where ν ∈ [0, 1]. Here we stick to the conventional notation for the geometric mean. Moreover, Raissouli, Moslehian and Furuichi (see [12, Proposition 2.1]) proved that if A, B are positive definite, then sin νπ π
∞
s ν−1 (A−1 + s B −1 )−1 ds = A1/2 (A−1/2 B A−1/2 )ν A1/2 ,
0
that is, (1.2) is a natural extension of the geometric mean for positive definite matrices. In this paper, we devote to the study of the inequalities for the Heinz mean for sector matrices. Let A, B ∈ Mn with W (A), W (B) ⊂ Sα . We define the Heinz mean of sector matrices (in particular, positive definite matrices) to be as follows: Hν (A, B)
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