More on operator monotone and operator convex functions of several variables

Let C1,C2,…,Ck be positive matrices in Mn and f be a continuous real-valued function on [0,∞). In addition, consider Φ as a positive linear functional on Mn and defineϕ(t1,t2,t3,…,tk)=Φ(f(t1C1+t2C2+t3C3+…+tkCk)), as a k variables continuous function on [0,∞)×…×[0,∞). In this paper, we show that if f is an operator convex function of order mn, then ϕ is a k variables operator convex function of order (n1,…,nk) such that m=n1n2…nk. Also, if f is an operator monotone function of order nk+1, then ϕ is a k variables operator monotone function of order n. In particular, if f is a non-negative operator decreasing function on [0,∞), then the function t→Φ(f(A+tB)) is an operator decreasing and can be written as a Laplace transform of a positive measure.