Operator inequality implying generalized Bebiano–Lemos–Providência one Original Research Article

A capital letter means n×n matrix. Very recently, Fujii et al. show: For every A,Bgreater-or-equal, slanted0 and pgreater-or-equal, slanted1">imageholds for any sgreater-or-equal, slanted0. In fact, (star, filled) yields Bebiano–Lemos–Providência inequality image for sgreater-or-equal, slantedtgreater-or-equal, slanted0. As an extension of (star, filled), we show the following result. The following (i) and (ii) hold and they are equivalent: (i) For every A>0, Bgreater-or-equal, slanted0, 0less-than-or-equals, slantαless-than-or-equals, slant1 and each tset membership, variant[0,1], and any real number q≠0image holds for sgreater-or-equal, slanted1 and rgreater-or-equal, slantedt, where image and image. (ii) . If Agreater-or-equal, slantedBgreater-or-equal, slanted0 with A>0, then for tset membership, variant[0,1] and pgreater-or-equal, slanted1image for sgreater-or-equal, slanted1 and rgreater-or-equal, slantedt. (i) implies (star, filled). In fact, put t=0, s=1, qr=1 in (i), then image image holds, and finally, replace B by Bp+s, A by As, α by image and also replace q by image, then we have the desired (star, filled).