# logarithmically convex function

###### Definition.

A function^{} $f:[a,b]\to \mathbb{R}$ such that $f(x)>0$ for all $x$ is said
to be logarithmically convex if $\mathrm{log}f(x)$ is a convex function.

It is easy to see that a logarithmically convex function is a convex function, but the converse is not true. For example $f(x)={x}^{2}$ is a convex function, but $\mathrm{log}f(x)=\mathrm{log}{x}^{2}=2\mathrm{log}x$ is not a convex function and thus $f(x)={x}^{2}$ is not logarithmically convex. On the other hand ${e}^{{x}^{2}}$ is logarithmically convex since $\mathrm{log}{e}^{{x}^{2}}={x}^{2}$ is convex. A less trivial example of a logarithmically convex function is the gamma function^{}, if restricted to the positive reals.

The definition is easily extended to functions $f:U\subset \mathbb{R}\to \mathbb{R}$, for any connected set $U$ (where still we have $f>0$), in the obvious way. Such a function is logarithmically convex if it is logarithmically convex on all intervals $[a,b]\subset U$.

## References

- 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.

Title | logarithmically convex function |

Canonical name | LogarithmicallyConvexFunction |

Date of creation | 2013-03-22 14:13:33 |

Last modified on | 2013-03-22 14:13:33 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 7 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 26A51 |

Synonym | logarithmically convex |

Synonym | log-convex function |

Synonym | log-convex |

Synonym | log convex function |

Synonym | log convex |

Related topic | ConvexFunction |

Related topic | BohrMollerupTheorem |

Related topic | HadamardThreeCircleTheorem |