An order preserving inequality via Furuta inequality, II

Furuta showed that if A B>0, then for each r 0, F(p)=(Br/2ApBr/2)(1+r)/(p+r) is increasing for p 1. But he pointed out that this result does not remain valid for 0 p 1 and r 0. In this paper, a necessary and sufficient condition for (Br/2Aα1Br/2)β/(α1+r) (Br/2Aα2Br/2)β/(α2+r) is obtained, and by using this comparison technique, we show some operator inequalities about the monotonicity of p [0,1] under some conditions of exponents.