"In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906,[1] building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Hölder in 1889.[2] Given its generality, the inequality appears in many forms depending on the context, some of which are presented below. In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation; it is a simple corollary that the opposite is true of concave transformations"
If you connect any two points, it lies outside the curve. Which is basically the intuition for Jensen's inequality: if you go partway between two points it's above the curve, so the weighted average of the curve at those two points is bigger than the curve at their weighted average.
A convex function is a function that is bowl shaped such a parabola, `x^2`. If you take two points and connect them with a straight line, then Jensen's inequality tells you that the function lies below this straight line. Basically, `f(cx+(1-c)y) <= c f(x) + (1-c) f(y)` for `0<=c<=1`. The expression `cx+(1-c)y` provides a way to move between a point `x` and a point `y`. The expression on the left of the inequality is the evaluation of the function along this line. The expression on the right is the straight line connecting the two points.
There are a bunch of generalizations to this. It works for any convex combination of points. A convex combination of points is a weighted sum of points where the weights are positive and add to 1. If one is careful, eventually this can become an infinite convex combination of points, which means that the inequality holds with integrals.
In my opinion, the wiki article is not well written.
Goes to wikipedia
"In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906,[1] building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Hölder in 1889.[2] Given its generality, the inequality appears in many forms depending on the context, some of which are presented below. In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation; it is a simple corollary that the opposite is true of concave transformations"
I still have no idea what it means.