Generalizations of reverse Bebiano–Lemos–Providência inequality

In our recent paper, we generalized Bebiano–Lemos–Providência inequality (BLP inequality) that for A,B⩾0A,B⩾0‖A1+t2BtA1+t2‖⩽‖A12(As2BsAs2)tsA12‖for all s⩾t⩾0s⩾t⩾0. On the other hand, we also propose a reverse of BLP inequality, which is inspired by Araki–Cordes inequality; for A,B>0A,B>0‖A1+t2BtA1+t2‖⩾‖A12As2BsAs2tsA12‖for all t⩾s⩾1t⩾s⩾1.Based on our results, we discuss the reverse of BLP inequality in a general setting, in which we point out that the restriction t⩾s⩾1t⩾s⩾1 in the above is quite reasonable.