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 .
Estimations of Heron means for positive operators
The arithmetic-geometric mean inequality induces the path of Heron means through these two means by H r (A,B) = r(A μB)+(1− r)(A∇μB) for each μ ∈ [0,1] , r ∈ R and positive operators A, B on a Hilbert space. In this note, we estimate H r (A,B) by the harmonic mean. As an application of this method, we refine the arithmetic-geometric mean inequality under the assumption of the strict order A−B m > 0 .
Geometrical significance of the Löwner-Heinz inequality
It is proven that the Lowner-Heinz inequality ‖AtBt‖ ≤ ‖AB‖t, valid for all positive invertible operators A,B on the Hilbert space H and t ∈ [0, 1], has equivalent forms related to the Finsler structure of the space of positive invertible elements of L(H) or, more generally, of a unital C∗algebra. In particular, the Lowner-Heinz inequality is equivalent to some type of “nonpositive curvature” property of that space.