A Characterization of Convex Functions and Its Application to Operator Monotone Functions
نویسندگان
چکیده
We give a characterization of convex functions in terms of difference 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,∞), then f(A)−f(B) ≥ f(‖B‖+ )−f(‖B‖) > 0, where = ‖(A−B)−1‖−1. Moreover it gives a simple proof to Furuta’s theorem: If logA > logB for A, B > 0 and f is operator monotone on (0,∞), then there exists a β > 0 such that f(A) > f(B) for all 0 < α ≤ β.
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