Loewner matrices of matrix convex and monotone functions. (English) Zbl 1261.15026

Under some conditions on the real \(C^{1}\) functions \(f(t), tf(t)\), and \(t^{2}f(t)\) on \((0,\infty )\), the author obtains characterizations of matrix convexity and matrix monotony of the above functions in terms of the conditional negative or positive definiteness of Loewner matrices. The obtained results are applied to power functions \(t^{\alpha }\) on \((0,\infty )\). Similar characterizations of matrix monotone functions on a finite interval \((a,b)\) are obtained by utilizing an operator monotone bijection between\((a,b)\) and \((0,1)\).


15A45 Miscellaneous inequalities involving matrices
47A63 Linear operator inequalities
42A82 Positive definite functions in one variable harmonic analysis
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