Journal of the Mathematical Society of Japan

Loewner matrices of matrix convex and monotone functions

Fumio HIAI and Takashi SANO

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The matrix convexity and the matrix monotony of a real C1 function f on (0,∞) are characterized in terms of the conditional negative or positive definiteness of the Loewner matrices associated with f, tf(t), and t2f(t). Similar characterizations are also obtained for matrix monotone functions on a finite interval (a,b).

Article information

J. Math. Soc. Japan, Volume 64, Number 2 (2012), 343-364.

First available in Project Euclid: 26 April 2012

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Zentralblatt MATH identifier

Primary: 15A45: Miscellaneous inequalities involving matrices
Secondary: 47A63: Operator inequalities 42A82: Positive definite functions

n-convex n-monotone operator monotone operator convex Loewner matrix conditional negative definite conditional positive definite


HIAI, Fumio; SANO, Takashi. Loewner matrices of matrix convex and monotone functions. J. Math. Soc. Japan 64 (2012), no. 2, 343--364. doi:10.2969/jmsj/06420343.

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