The Characteristics Matrix Convex Functions

We can study the characterizations of matrix convex functions from two aspects. One is the function itself, the other is matrix inequality. Kraus is the precursor of the first aspect, who studied the matrix convexity of a function through investigating a matrix is positive semidefinite or not, and the elements of the matrix are the divided differences of the function. It's a very useful result, but when the order of the matrix is too large, it is very difficult to compute the divided differences. Hansen improved Kraus'result. He replaced the divided differences as derivatives in the matrix. And the matrix convexity of the function is related to the positive definiteness of the matrix. Using Hansen's result, we get that"a polynomial is n-convex, then its order has to be bigger than 2n". We also studied the characterizations of 2 -matrix convex functions using an inequality of divided functions.There are plentiful components about matrix inequalities, and they are used in many branches of mathematics. But only the inequalities about Hermite matrix have relation with matrix convex functions. Loewner and Kwong have done a lot of work about these. They emphasized on the matrices of p powers where p is positive, so these results are not helpful for constructing matrix convex functions. Bhagwat and Subramanian got the AM-HM inequality for matrices, and it's very helpful for the study of matrix convex functions. In this article we use the property of matrix monotone function, give a new proof of the inequality for matrix, and discuss when it is an equation. We prove that matrix convexity and matrix weak convexity are equivalent when the function is continuous, so the inverse proportion function is a matrix convex function.