Operator log-convex functions and operator means. (English) Zbl 1221.47028
Summary: We study operator log-convex functions on \((0, \infty )\) and prove that a continuous nonnegative function on \((0, \infty )\) is operator log-convex if and only if it is operator monotone decreasing. Several equivalent conditions related to operator means are given for such functions. Operator log-concave functions are also discussed.
MSC:
| 47A63 | Linear operator inequalities |
| 47A64 | Operator means involving linear operators, shorted linear operators, etc. |
| 15A45 | Miscellaneous inequalities involving matrices |
Keywords:
operator log-convex functions; operator monotone decreasing; operator means; operator log-concave functionsReferences:
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