# Jensen’s inequality

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 ${\mu}_{k}\ge 0$ one has:

$$f\left(\frac{{\sum}_{k=1}^{n}{\mu}_{k}{x}_{k}}{{\sum}_{k}^{n}{\mu}_{k}}\right)\le \frac{{\sum}_{k=1}^{n}{\mu}_{k}f\left({x}_{k}\right)}{{\sum}_{k}^{n}{\mu}_{k}}.$$ |

A common situation occurs when ${\mu}_{1}+{\mu}_{2}+\mathrm{\cdots}+{\mu}_{n}=1$; in this case, the inequality^{} simplifies to:

$$f\left(\sum _{k=1}^{n}{\mu}_{k}{x}_{k}\right)\le \sum _{k=1}^{n}{\mu}_{k}f({x}_{k})$$ |

where $0\le {\mu}_{k}\le 1$.

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

Example:

$f\mathbf{}\mathrm{(}x\mathrm{)}\mathrm{=}{x}^{\mathrm{2}}$ is a convex function on $[0,10]$.
Then

$${(0.2\cdot 4+0.5\cdot 3+0.3\cdot 7)}^{2}\le 0.2({4}^{2})+0.5({3}^{2})+0.3({7}^{2}).$$ |

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

$$f\left(\frac{1}{n}\sum _{k=1}^{n}{x}_{k}\right)\le \frac{1}{n}\sum _{k=1}^{n}f({x}_{k})$$ |

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 $X$:

$$\mathrm{E}[f(X)]\ge f(\mathrm{E}[X]).$$ |

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

Title | Jensen’s inequality |

Canonical name | JensensInequality |

Date of creation | 2013-03-22 11:46:30 |

Last modified on | 2013-03-22 11:46:30 |

Owner | Andrea Ambrosio (7332) |

Last modified by | Andrea Ambrosio (7332) |

Numerical id | 13 |

Author | Andrea Ambrosio (7332) |

Entry type | Theorem |

Classification | msc 81Q30 |

Classification | msc 26D15 |

Classification | msc 39B62 |

Classification | msc 18-00 |

Related topic | ConvexFunction |

Related topic | ConcaveFunction |

Related topic | ArithmeticGeometricMeansInequality |

Related topic | ProofOfGeneralMeansInequality |