Jensen's Inequality (Real Analysis)

Theorem
Let $I$ be a real interval.

Let $\phi : I \to \R$ be a convex function.

Let $x_1, x_2, \ldots, x_n \in I$.

Let $\lambda_1, \lambda_2, \ldots, \lambda_n \ge 0$ be real numbers, at least one of which is non-zero.

Then:
 * $\displaystyle \phi\left(\frac {\sum_{k=1}^n \lambda_k x_k} {\sum_{k=1}^n \lambda_k}\right) \le \frac {\sum_{k=1}^n \lambda_k \phi\left({x_k}\right) } {\sum_{k=1}^n \lambda_k}$

Also, if $\phi$ is strictly convex, then equality holds if and only if $x_1 = x_2 = \cdots = x_n$.

Proof
We proceed by mathematical induction on $n$.

For $n = 1$, the statement is trivial.

Now, assume that the theorem holds for some value of $n$.

We will show that the theorem then holds for the value $n + 1$.

Assume without loss of generality that $\lambda_n$ and $\lambda_{n+1}$ are non-zero. Otherwise, the statement reduces to the induction hypothesis.

Define $\displaystyle y = \frac {\lambda_n x_n + \lambda_{n+1} x_{n+1}} {\lambda_n + \lambda_{n+1}}$.

Then:

Note that in the case that $\phi$ is strictly convex, equality holds if and only if $x_1 = x_2 = \cdots = x_n = x_{n+1}$.