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

There are basic equivalent assertions known for operator monotone functions and operator convex functions in two papers by Hansen and Pedersen. In this note we consider their results as correlation problem between two sequences of matrix nn-monotone functions and matrix nn-convex functions, and we focus the following three assertions at each label nn among them:(i)f(0)⩽0f(0)⩽0 and ff is nn-convex in [0,α)[0,α),(ii)For each matrix aa with its spectrum in [0,α)[0,α) and a contraction cc in the matrix algebra MnMn,f(cac)⩽cf(a)c,f(cac)⩽cf(a)c,(iii)The function f(t)/t(=g(t)) is nn-monotone in (0,α)(0,α).We show that for any n∈Nn∈N two conditions (ii) and (iii) are equivalent. The assertion that ff is nn-convex with f(0)⩽0f(0)⩽0 implies that g(t)g(t) is (n-1)(n-1)-monotone holds. The implication from (iii) to (i) does not hold even for n=1n=1. We also show in a limited case that the condition (i) implies (ii).