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).
"Loewner matrices of matrix convex and monotone functions." J. Math. Soc. Japan 64 (2) 343 - 364, April, 2012. https://doi.org/10.2969/jmsj/06420343