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Jensen's inequality (Theorem)

If $f$ is a convex function on the interval $[a,b]$, for each $\left\{x_k\right\}_{k=1}^n \in[a,b]$ and each $\left\{\mu_k\right\}_{k=1}^n$ with $\mu_{k}\geq0$ one has:

\begin{displaymath}f\left(\frac{\sum_{k=1}^{n}\mu_{k}x_{k}}{\sum_{k}^{n}\mu_{k}}... ...\sum_{k=1}^{n}\mu_{k}f\left(x_{k}\right)}{\sum_{k}^{n}\mu_{k}}.\end{displaymath}

A common situation occurs when $\mu_1+\mu_2+\cdots+\mu_n=1$; in this case, the inequality simplifies to:


\begin{displaymath}f\left(\sum_{k=1}^n \mu_k x_k\right)\leq \sum_{k=1}^n \mu_k f(x_k)\end{displaymath}

where $0\le \mu_k\le 1$.

If $f$ is a concave function, the inequality is reversed.


Example:
$f(x)=x^2$ is a convex function on $[0,10]$. Then

\begin{displaymath}(0.2\cdot4+ 0.5\cdot3+0.3\cdot7)^2 \leq 0.2(4^2) + 0.5(3^2)+0.3(7^2).\end{displaymath}


A very special case of this inequality is when $\mu_k=\frac{1}{n}$ because then

\begin{displaymath}f\left(\frac{1}{n}\sum_{k=1}^n x_k\right)\le\frac{1}{n}\sum_{k=1}^n f(x_k)\end{displaymath}

that is, the value of the function at the mean of the $x_k$ is less or equal than the mean of the values of the function at each $x_k$.

There is another formulation of Jensen's inequality used in probability:
Let $X$ be some random variable, and let $f(x)$ be a convex function (defined at least on a segment containing the range of $X$). Then the expected value of $f(X)$ is at least the value of $f$ at the mean of $X$:

\begin{displaymath} \mathrm{E}[f(X)] \ge f(\mathrm{E}[ X]). \end{displaymath}

With this approach, the weights of the first form can be seen as probabilities.



"Jensen's inequality" is owned by Andrea Ambrosio. [ full author list (2) | owner history (2) ]
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See Also: convex function, concave function, arithmetic-geometric-harmonic means inequality, proof of general means inequality

Keywords:  Convex, Concave, Inequality

Attachments:
proof of arithmetic-geometric-harmonic means inequality (Example) by mathcam
proof of Jensen's inequality (Proof) by Andrea Ambrosio
another proof of Jensen's inequality (Proof) by Andrea Ambrosio
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Cross-references: weights, expected value, range, segment, random variable, mean, function, concave function, inequality, interval, convex function
There are 7 references to this entry.

This is version 8 of Jensen's inequality, born on 2001-10-15, modified 2006-09-13.
Object id is 234, canonical name is JensensInequality.
Accessed 59074 times total.

Classification:
AMS MSC: 39B62 (Difference and functional equations :: Functional equations and inequalities :: Functional inequalities, including subadditivity, convexity, etc.)
 26D15 (Real functions :: Inequalities :: Inequalities for sums, series and integrals)
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