Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4603572 | Linear Algebra and its Applications | 2008 | 11 Pages |
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.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Takayuki Furuta,