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Hiai, Fumio; Sano, Takashi

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

J. Math. Soc. Japan 64, No. 2, 343-364 (2012).
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 \((0,1)\).
Reviewer: Fozi Dannan (Damascus)

Cited in 9 Documents

MSC:

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

Keywords:

\(n\)-convex; \(n\)-monotone; operator convex; Loewner matrix; conditional negative definite; conditional positive definite; matrix convexity; matrix monotony
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\textit{F. Hiai} and \textit{T. Sano}, J. Math. Soc. Japan 64, No. 2, 343--364 (2012; Zbl 1261.15026)
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