Operator inequalities: From a general theorem to concrete inequalities

The aim of this paper is to give a method to extract concrete inequalities from a general theorem, which is established by making use of majorization relation between functions. By this method we can get a lot of inequalities; among others we extend Furuta inequality as follows: Let fifi, gjgj be positive operator monotone functions on [0,∞)[0,∞) and put k(t)=tr0f1(t)r1⋯fm(t)rmk(t)=tr0f1(t)r1⋯fm(t)rm, h(t)=tp0g1(t)p1⋯gn(t)pnh(t)=tp0g1(t)p1⋯gn(t)pn, where p0≥1p0≥1 and ri≥0ri≥0, pj≥0pj≥0. Then 0≤A≤C≤B0≤A≤C≤B implies, for 0<α≤1+r0p+r0 with p=p0+⋯+pnp=p0+⋯+pn, (k(C)12h(A)k(C)12)α≤(k(C)12h(C)k(C)12)α≤(k(C)12h(B)k(C)12)α. Moreover, we show log⁡C1/2eAC1/2≤log⁡C1/2eCC1/2≤log⁡C1/2eBC1/2log⁡C1/2eAC1/2≤log⁡C1/2eCC1/2≤log⁡C1/2eBC1/2, provided C is invertible. We also refer to operator geometric mean.