Monotonicity of order preserving operator functions

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.