If a function is convex on an interval, then Jensen's inequality holds, which says for any number of points in the interval, the average value of the function values at these points is greater than or equal to the function value of the average value of the points: average of f(x) ≧ f(average of x_i).
I first explain what a convex function means, and then, give a proof of Jensen's inequality by using mathematical induction.