# Landau’s constant

We suggest that the reader reads first the entry on Bloch’s constant. Let $\mathcal{F}$ be the set of all functions $f$ holomorphic on a region containing the
closure of the disk $$ and satisfying
$f(0)=0$ and ${f}^{\prime}(0)=1$. For each $f\in \mathcal{F}$ let $\lambda (f)$
be the supremum of all numbers $r$ such that there is a disk
$S\subset D$ such that $f(S)$ contains a disk of radius $r$ (notice that here we don’t require $f$ to be injective^{} on $S$).

###### Definition.

Landau’s constant $L$ is defined by

$$L=inf\{\lambda (f):f\in \mathcal{F}\}.$$ |

Let $B$ be Bloch’s constant. Then, clearly, $L\ge B$. The exact value of $L$ (as that of $B$) is not known but it has been shown that $0.5\le L\le 0.56$. In particular, it is known that $L$ is strictly greater than $B$.

## References

- 1 John B. Conway, Functions of One Complex Variable I, Second Edition, 1978, Springer-Verlag, New York.

Title | Landau’s constant |
---|---|

Canonical name | LandausConstant |

Date of creation | 2013-03-22 15:58:07 |

Last modified on | 2013-03-22 15:58:07 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 4 |

Author | alozano (2414) |

Entry type | Definition |

Classification | msc 32H02 |

Related topic | BlochsConstant |