Source: J. Math. Soc. Japan
Volume 64, Number 2
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).
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R. B. Bapat and T. E. S. Raghavan, Nonnegative Matrices and Applications, Cambridge University Press, Cambridge, 1997.
R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1996.
R. Bhatia and T. Sano, Loewner matrices and operator convexity, Math. Ann., 344 (2009), 703–716.
R. Bhatia and T. Sano, Positivity and conditional positivity of Loewner matrices, Positivity, 14 (2010), 421–430.
A. M. Bruckner and E. Ostrow, Some function classes related to the class of convex functions, Pacific J. Math., 12 (1962), 1203–1215.
Mathematical Reviews (MathSciNet): MR148822
W. F. Donoghue, Jr., Monotone Matrix Functions and Analytic Continuation, Springer-Verlag, Berlin-Heidelberg-New York, 1974.
Mathematical Reviews (MathSciNet): MR486556
F. Hansen and G. K. Pedersen, Jensen's inequality for operators and Löwner's theorem, Math. Ann., 258 (1982), 229–241.
F. Hansen and J. Tomiyama, Differential analysis of matrix convex functions II, J. Inequal. Pure Appl. Math., 10 (2009), Article 32, 5 pp.
F. Hiai, Matrix analysis: matrix monotone functions, matrix means, and majorization, Interdiscip. Inform. Sci., 16 (2010), 139–248.
R. A. Horn, Schlicht mappings and infinitely divisible kernels, Pacific J. Math., 38 (1971), 423–430.
Mathematical Reviews (MathSciNet): MR310208
F. Kraus, Über konvexe Matrixfunktionen, Math. Z., 41 (1936), 18–42.
K. Löwner, Über monotone Matrixfunktionen, Math. Z., 38 (1934), 177–216.
H. Osaka and J. Tomiyama, Double piling structure of matrix monotone functions and of matrix convex functions, Linear Algebra Appl., 431 (2009), 1825–1832.
M. Uchiyama, Operator monotone functions, positive definite kernels and majorization, Proc. Amer. Math. Soc., 138 (2010), 3985–3996.