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On the convexity of the Heinz means. (English) Zbl 1230.47026

The basic property of convex functions plays a central role in this paper, that is, the Hermite-Hadamard integral inequality for convex functions.
Let \(A,B,X\) be operators on a complex separable Hilbert space such that \(A\) and \(B\) are positive, and let \(0 \leq v \leq 1\). The Heinz inequality asserts that, for every unitarily invariant norm \(|||\cdot|||\), \[ 2|||A^{1/2}XB^{1/2}||| \leq ||| A^{v}XB^{1-v}+A^{1-v}XB^{v} ||| ||| AX+XB |||. \] Using the convexity of the function \(f(v)= ||| A^{v}XB^{1-v}+A^{1-v}XB^{v} |||\) on \([0,1]\), the author obtains several refinements of these norm inequalities and investigates their equality conditions.

MSC:

47A30 Norms (inequalities, more than one norm, etc.) of linear operators
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
26D15 Inequalities for sums, series and integrals
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