Question

Problem 2
Find all functions F(2): R to R
having the property that for any X1 and X2 the following inequality holds:

$f(x1) - f(x2) \leqslant (x1 - x2) {}^{2}$
​

Answer 1

Answer:

$\huge \colorbox{lime}{Answer}$

(f-f)(x1-x2)

Answer 2

If we define g(x)=x2, we can write the above inequality as E[g(X)]≥g(E[X]). The function g(x)=x2 is an example of convex function. Jensen's inequality states that, for any convex function g, we have E[g(X)]≥g(E[X]).

(f-f) (x1-x2)

Jensen's Rule

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.

Jensen's Inequality is a useful tool in mathematics, specifically in applied fields such as probability and statistics. For example, it is often used as a tool in mathematical proofs. It is also used to make claims about a function where little is known or needs to be known about the distribution.

Jensen's inequality states that this line is everywhere at least as large as f(x). pf(x1) + (1 − p)f(x2) ≥ f(px1 + (1 − p)x2). If f is (doubly) differentiable then f is convex if and only if d2f/dx2 ≥ 0. Now consider a probability distribution P on a set M and a function X assigning real values X(m) for m ∈ M.

The function g(x)=x2 is an example of convex function. Jensen's inequality states that, for any convex function g, we have E[g(X)]≥g(E[X]). ... On the other hand, if the line segment always lies below the graph, the function is said to be concave.