Generalized Bebiano–Lemos–Providência inequalities and their reverses

We improve Bebiano-Lemos-Providência inequality: For A,B⩾0A,B⩾0A1+t2BtA1+t2⩽A12As2BsAs2tsA12for all s⩾t⩾0s⩾t⩾0. In our discussion, Furuta inequality plays an essential role. Actually we haveA1+s2B1+sA1+s2p+sp(1+s)⩽A12As2Bp+sAs21pA12for p⩾1p⩾1 and s⩾0s⩾0.