Support functions and analytic inequalities (Q1076826)

Let K be a convex cone in the Euclidean n-dimensional space \(R^ n\), and let \(f: K\to R\) be a convex positively homogeneous function. Then for all vectors x,y\(\in K\) the following inequalities are valid: \[ f(x+y)\leq f(x)+f(y),\quad f'(y;x)\leq f(x). \] By using this result the author derives some classical inequalities: the Minkowski and Hölder inequalities, the arithmetic-geometric mean inequality, inequalities for elementary symmetric functions, determinantal inequalities, and inequalities for indefinite forms.