Refinements of a reversed AM–GM operator inequality

Abstract

We prove some refinements of a reverse AM-GM operator inequality due to M. Lin [Studia Math. 2013;215:187-194]. In particular, we show the operator inequalitywhere A, B are positive operators on a Hilbert space such that 0 < m ≤ A, B ≤ M for some positive numbers m, M , Φ is a positive unital linear map, ν ∈ [0, 1], r = min{ν, 1 − ν}, p > 0 and α = max (M+m) 2 4Mm , (M+m) 2 4 2 p Mm . 2010 Mathematics Subject Classification. Primary 47A63, Secondary 47A60. Key words and phrases. the operator arithmetic mean, the operator geometric mean, the operator harmonic mean, positive unital linear map, reverse AM-GM operator inequality.

New generalized inequalities using arbitrary operator means and their duals

“…Remark 3.2. By putting α = 2, µ = 1 2 and taking q → 0, inequality (23) collapse to the derived result in [2].…”

Improvements of some operator inequalities involving positive linear maps via the Kantorovich constant

“…In this article, we give some operator inequalities involving positive linear maps that generalize inequalities (1.8), (1.9) and refine some results in [2,17]. Moreover, we obtain a reverse of Ando's inequality.…”

“…For further information about the AM-GM operator inequality and positive linear maps inequalities we refer the reader to [1,2,3,8,14] and references therein. Lin [11] presented a reverse of inequality (1.1) for a positive linear map Φ and positive operators A, B ∈ B(H ) such that m ≤ A, B ≤ M as follows:…”

Further refinements of reversed AM–GM operator inequalities

“…The Kantorovich constant is defined by K(t, 2) = (t+1) 2 4t for t > 0. What's more, the relative operator entropy of A and B is defined as S(A|B) = A 1 2 log(A -1 2 BA -1 2 )A 1 2 . For A = (a ij ) ∈ M n , the Hilbert-Schmidt norm is defined by A 2 = n i,j=1 a 2 ij .…”

“…What's more, the relative operator entropy of A and B is defined as S(A|B) = A 1 2 log(A -1 2 BA -1 2 )A 1 2 . For A = (a ij ) ∈ M n , the Hilbert-Schmidt norm is defined by A 2 = n i,j=1 a 2 ij . As we all know that • 2 has the unitary invariance property: UAV 2 = A 2 for all A ∈ M n and unitary matrices U, V ∈ M n .…”