Some New Methods for Generating Convex Functions

We present some new methods for constructing convex functions. One of the methods is based on the composition of a convex function of several variables which is separately monotone with convex and concave functions. Using several well-known results on the composition of convex and quasi-convex functions we build new convex, quasi-convex, concave, and quasi-concave functions. The third section is dedicated to the study of convexity property of symmetric Archimedean functions. In the fourth section the asymmetric Archimedean function is considered. A classical example of such a function is the Bellman function. The fifth section is dedicated to the study of convexity/concavity of symmetric polynomials. In the sixth section a new proof of Chandler–Davis theorem is given. Starting from symmetric convex functions defined on finite dimensional spaces we build several convex functions of hermitian matrices. The seventh section is dedicated to a generalization of Muirhead’s theorem and to some applications of it. The last section is dedicated to the construction of convex functions based on Taylor remainder series.