A CHARACTERIZATION OF CONVEX FUNCTIONS AND ITS APPLICATION TO OPERATOR MONOTONE FUNCTIONS
We give a characterization of convex functions in terms of dier- ence among values of a function. As an application, we propose an estimation of operator monotone functions: If A > B 0 and f is operator monotone on (0,1), then f(A) f(B) f(kBk+ ) f(kBk) > 0, where = k(A B) 1 k 1 .
Some extensions of the operator entropy type inequalities
ABSTRACT In this paper, we define the generalised relative operator entropy and investigate some of its properties such as subadditivity and homogeneity. As application of our result, we obtain the information inequality. In continuation, we establish some reverses of the operator entropy inequalities under certain conditions by using the Mond–Pečarić method.
The best possibility of the grand Furuta inequality
. Let A,B 2 B ( H ) be invertible bounded linear operators on a Hilbert space H satisfying O (cid:20) B (cid:20) A , and let p,r,s,t be real numbers satisfying 1 < s, 0 < t < 1 ,t (cid:20) r, 1 (cid:20) p. Furuta showed that if 0 < α (cid:20) 1 − t + r ( p − t ) s + r , then n A r 2 (cid:16) A − t 2 B p A − t 2 (cid:17) s A r 2 o α (cid:20) A f ( p − t ) s + r g α . This inequality is called the grand Furuta inequality, which interpolates the Furuta inequality ( t = 0) and the Ando-Hiai inequality ( t = 1 ,r = s ). In this paper, we show the grand Furuta inequality is best possible in the following sense: that is, if 1 − t + r ( p − t ) s + r < α , then there exist invertible matrices A,B with O (cid:20) B (cid:20) A which do not satisfy n A r 2 (cid:16) A − t 2 B p A − t 2 (cid:17) s A r 2 o α (cid:20) A f ( p − t ) s + r g α .