Log majorization via an order preserving operator inequality

A capital letter designates an n×nn×n matrix. For every  A>0A>0, B⩾0B⩾0, 0⩽α⩽10⩽α⩽1and each  t∈[0,1],(A♯αB)h≻(log)t∈[0,1],(A♯αB)h≻(log)A1-t+r♯β(A1-t♮sB)A1-t+r♯β(A1-t♮sB)holds for  s⩾1s⩾1and  r⩾tr⩾t, where  h=(1-t+r)s(1-αt)s+αrand  β=hsα and this result is an extension of the following one by Ando-Hiai: (A♯αB)r≻(log)Ar♯αBr(A♯αB)r≻(log)Ar♯αBrfor  r⩾1.r⩾1. From this point of view, we shall show the following further extension. For every  A>0A>0, B⩾0B⩾0, t∈[0,1]t∈[0,1] and p1,p2,….,p2(n-1),p2n-1,p2n⩾1p1,p2,….,p2(n-1),p2n-1,p2n⩾1for any natural number  nnand  r⩾tr⩾t,(A♯1p1B)h≻(log)A1-t+r♯β{A1-t♮p2n{A♮p2n-1{A1-t♮p2n-2{A♮p2n-3{A1-t…{A♮p3(A1-t♮p2B)}}….}}︷ntimes︸A1-tappearsntimesandAappearn-1timesbyturnsholds, where  hh, ββand  φ[2n;r,t]φ[2n;r,t]are as follows:h=p1p2…p2n(1-t+r)φ[2n;r,t]andβ=hp1p2……p2n,where  φ[2n;r,t]={…..[{[(p1-t)p2+t]p3-t}p4+t]p5-……-t}p2n+r︸-tappearsntimesandtappearsn-1timesbyturns=r+∏i=12npi+(∏i=32npi+∏i=52npi+…….+∏i=72npi+…+p2n-1p2n︸n-1terms)t-(∏i=22npi+∏i=42npi+∏i=62npi+…….+p2(n-1)p2n-1p2n+p2n︸nterms)t.