Refinements of the heinz inequalities for operators and matrices

Mahdi Mohammadi Gohari; Maryam Amyari

doi:10.1515/ms-2017-0192

Article

Refinements of the heinz inequalities for operators and matrices

Mahdi Mohammadi Gohari and Maryam Amyari

Published/Copyright: November 20, 2018

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From the journal Mathematica Slovaca Volume 68 Issue 6

Abstract

Suppose that A, B ∈ 𝔹(𝓗) are positive invertible operators. In this paper, we show that

A#B≤11−2μA12Fμ(A−12BA−12)A12 ≤12[A#B+Hμ(A,B)] ≤12[11−2μA12Fμ(A−12BA−12)A12+Hμ(A,B)] ≤⋯≤12nA#B+2n−12nHμ(A,B) ≤12n(1−2μ)A12Fμ(A−12BA−12)A12+2n−12nHμ(A,B) ≤12n+1A#B+2n+1−12n+1Hμ(A,B) ≤⋯≤Hμ(A,B)

for each μ∈[0,1]∖{12}, where H_μ (A, B) and A#B are the Heinz mean and the geometric mean for operators A, B, respectively, and Fμ∈C(sp(A−12BA−12)) is a certain parameterized class of functions. As an application, we present several inequalities for unitarily invariant norms.

MSC 2010: Primary 47A63; 15A60; Secondary 26A51

Keywords: convex function; Heinz operator inequalities; Heinz mean; Hermite-Hadamard inequality; refinement; unitarily invariant norm

amyari@mshdiau.ac.ir

(Communicated by Werner Timmermann)

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Received: 2017-06-20

Accepted: 2017-09-14

Published Online: 2018-11-20

Published in Print: 2018-12-19

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https://doi.org/10.1515/ms-2017-0192

Keywords for this article

convex function; Heinz operator inequalities; Heinz mean; Hermite-Hadamard inequality; refinement; unitarily invariant norm