Convex functions and inequalities in the secondary school

Abstract

A function f : Df → R is called convex if for any x, y ∈ Df the inequality f ( x+y/2 ) \leq (f (x)+f (y) )/2 holds. We prove the main property of the convex fonctions (inequality (4)) and also the inequalities which the arithmetic, geometric, harmonic and quadratic means of several positive numbers satisfy. Then we prove some trigonometric and geometric inequalities.

Keywords

• function
• convex function
• arithmetic
• geometric
• harmonic and quadraticmeans
• trigonometric and geometric inequalities