Operator inequality implying generalized Bebiano–Lemos–Providência one

A capital letter means n×nn×n matrix. Very recently, Fujii et al. show: For every A,B⩾0A,B⩾0 and p⩾1p⩾1equation(★)A12As2Bp+sAs21pA12p(1+s)p+s≻(log)A1+s2B1+sA1+s2holds for any s⩾0s⩾0.In fact, (★) yields Bebiano–Lemos–Providência inequality A12(As2BsAs2)tsA12≻(log)A1+t2BtA1+t2 for s⩾t⩾0s⩾t⩾0. As an extension of (★), we show the following result. The following (i) and (ii) hold and they are equivalent:(i)For every A>0A>0, B⩾0B⩾0, 0⩽α⩽10⩽α⩽1 and each t∈[0,1]t∈[0,1], and any real number q≠0q≠0Aq2A12BA12αAq2h≻(log)Aq(1-t+r)2A-qr2A1+qt2BA1+qt2sA-qr2βAq(1-t+r)2 holds for s⩾1s⩾1 and r⩾tr⩾t, where β=α(1-t+r)(1-αt)s+αr and h=(1-t+r)s(1-αt)s+αr.(ii). If A⩾B⩾0A⩾B⩾0 with A>0A>0, then for t∈[0,1]t∈[0,1] and p⩾1p⩾1A1-t+r⩾Ar2A-t2BpA-t2sAr21-t+r(p-t)s+r for s⩾1s⩾1 and r⩾tr⩾t.(i) implies (★). In fact, put t=0t=0, s=1s=1, qr=1qr=1 in (i), then {Aq2(A12BA12)αAq2}1+qα+q≻(log)A1+q2Bα(1+q)α+qA1+q2 holds, and finally, replace B   by Bp+sBp+s, A   by AsAs, αα by 1p and also replace q   by 1s, then we have the desired (★).