Symmetric cone monotone functions and symmetric cone convex functions

Symmetric cone (SC) monotone functions and SC-convex functions are real scalar valued functions which induce Löwner operators associated with a simple Euclidean Jordan algebra to preserve the monotone order and convex order, respectively. In this paper, for a general simple Euclidean Jordan algebra except for octonion case, we show that the SC-monotonicity (respectively, SC-convexity) of order r is implied by the matrix monotonicity (respectively, matrix convexity) of some fixed order r' (≥ r). As a consequence, we draw the conclusion that (except for octonion case) a function is SC-monotone (respectively, SC-convex) if and only if it is matrix monotone (respectively, matrix convex).