Inequalities for Semi-convex Matrix Functions

ƒ defined on the set T of matrices with values in the set of matrices is said to be semi-convex if X, Y∈T, X≤Y implies that λX+(1−λ)Y∈T, 0<λ<1, and ƒ(λX+(1−λ)Y)≤λƒ(X)+(1−λ)ƒ(Y). Here X≤Y means that every element of Y−X is nonnegative. In this paper, a number of different inequalities are given for semi-convex functions of matrices.