Double piling structure of matrix monotone functions and of matrix convex functions II

We continue the analysis in [H. Osaka, J. Tomiyama, Double piling structure of matrix monotone functions and of matrix convex functions, Linear and its Applications 431 (2009) 1825–1832] in which the followings three assertions at each label n are discussed:(1)f(0)⩽0f(0)⩽0 and f is n  -convex in [0,α)[0,α)(2)For each matrix a   with its spectrum in [0,α)[0,α) and a contraction c   in the matrix algebra MnMn,f(c∗ac)⩽c∗f(a)c.f(c∗ac)⩽c∗f(a)c.(3)The function f(t)/tf(t)/t(=g(t))(=g(t)) is n  -monotone in (0,α)(0,α).We know that two conditions (2)(2) and (3)(3) are equivalent and if ff with f(0)≤0f(0)≤0 is nn-convex, then gg is (n-1)(n-1)-monotone. In this note we consider several extra conditions on ff or gg to conclude that the implication from (3)(3) to (1)(1) is true. In particular, we study a class Qn([0,α))Qn([0,α)) of functions with conditional positive Lowner matrix which contains the class of matrix nn-monotone functions and show that if f∈Qn+1([0,α))f∈Qn+1([0,α)) with f(0)=0f(0)=0 and gg is nn-monotone, then ff is nn-convex. We also discuss about the local property of nn-convexity.