Study On The Characteristics For Matrix Convex(Concave) Functions

As an important tool to study mathematics and its applications，matrix theory hasbeen widely used in mathematics and many scientific fields, such as mathematics,physics, economics, biology, management science and engineering, image analysis,automatic control image analysis etc. And the characterizations of matrix function(especially the convexity and monotonicity) is an important part of the matrixtheory.Traditionally,We can study the characterizations of matrix convex functionsfrom two aspects. One is the function itself, the other is matrix inequality. There areplentiful components about matrix inequalities, also they are used in many branches ofmathematics.But the inequalities related with matrix convex functions mainly isHermite matrix, so this thesis studies several matrix convex functions and its form ofJensen inequality from the latter that based on Hermite matrix.The main contents andinnovations are as follows:1. Studied on the Jensen inequalities and estimate the upper bound of the Jenseninequalities.2. Studied the reverse Jensen inequalities in the form of matrix convex functionand also expand the related conclusion.3.Based on operator mean theory,further studied the contact between the convexityand monotonicity of matrix function.