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

Abstract

We continue the analysis in [H. Osaka and 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 nn are discussed: (1)f(0)0f(0) \leq 0 and ff is nn-convex in [0,α)[0, \alpha). (2)For each matrix aa with its spectrum in [0,α)[0, \alpha) and a contraction cc in the matrix algebra MnM_n, f(cac)cf(a)cf(c^*ac) \leq c^*f(a)c. (3)The function f(t)/tf(t)/t (=g(t))(= g(t)) is nn-monotone in (0,α)(0, \alpha). We know that two conditions (2)(2) and (3)(3) are equivalent and if ff with f(0)0f(0) \leq 0 is nn-convex, then gg is (n1)(n -1)-monotone. In this note we consider several extra conditions on gg to conclude that the implication from (3)(3) to (1)(1) is true. In particular, we study a class Qn([0,α))Q_n([0, \alpha)) of functions with conditional positive Lowner matrix which contains the class of matrix nn-monotone functions and show that if fQn+1([0,α))f \in Q_{n+1}([0, \alpha)) 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.Comment: 13page

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