Article ID Journal Published Year Pages File Type
4603572 Linear Algebra and its Applications 2008 11 Pages PDF
Abstract

We discuss monotonicity of order preserving operator functions and related order preserving operator inequalities.Let  A⩾B⩾0A⩾B⩾0with  A>0,t∈[0,1]A>0,t∈[0,1]and  p⩾1p⩾1. LetF(λ,μ)=A-λ2Aλ2A-t2BpA-t2μAλ21-t+λ(p-t)μ+λA-λ2.We show that:(i)F(r,w)⩾F(r,1)⩾F(r,s)⩾F(r,s′)F(r,w)⩾F(r,1)⩾F(r,s)⩾F(r,s′)for any  s′⩾s⩾1,r⩾ts′⩾s⩾1,r⩾tand  1-tp-t⩽w⩽1,(ii)F(q,s)⩾F(t,s)⩾F(r,s)⩾F(r′,s)F(q,s)⩾F(t,s)⩾F(r,s)⩾F(r′,s)for any  r′⩾r⩾t,s⩾1r′⩾r⩾t,s⩾1and  t-1⩽q⩽tt-1⩽q⩽t.These imply the following recent inequality due to KameiAt♯1-tp-tBp⩾At2F(r,s)At2forr⩾tands⩾1.

Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory
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