World Library  
Flag as Inappropriate
Email this Article

Bloch's theorem (complex variables)

Article Id: WHEBN0011831990
Reproduction Date:

Title: Bloch's theorem (complex variables)  
Author: World Heritage Encyclopedia
Language: English
Subject: List of theorems, Bloch, Bloch's theorem, André Bloch (mathematician)
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Bloch's theorem (complex variables)

In complex analysis, a field within mathematics, Bloch's theorem gives a lower bound on the size of a disc in which an inverse to a holomorphic function exists. It is named after André Bloch.

Statement

Let f be a holomorphic function in the unit disk |z| ≤ 1. Suppose that |f′(0)| = 1. Then there exists a disc of radius b and an analytic function φ in this disc, such that f(φ(z)) = z for all z in this disc. Here b > 1/72 is an absolute constant.

Landau's theorem

If f is a holomorphic function in the unit disc with the property |f′(0)| = 1, then the image of f contains a disc of radius l, where lb is an absolute constant.

This theorem is named after Edmund Landau.

Valiron's theorem

Bloch's theorem was inspired by the following theorem of Georges Valiron:

Theorem. If f is a non-constant entire function then there exist discs D of arbitrarily large radius and analytic functions φ in D such that f(φ(z)) = z for z in D.

Bloch's theorem corresponds to Valiron's theorem via the so-called Bloch's Principle.

Bloch's and Landau's constants

The lower bound 1/72 in Bloch's theorem is not the best possible. The number B defined as the supremum of all b for which this theorem holds, is called the Bloch's constant. Bloch's theorem tells us B ≥ 1/72, but the exact value of B is still unknown.

The similarly defined optimal constant L in Landau's theorem is called the Landau's constant. Its exact value is also unknown.

The best known bounds for B at present are

\frac{\sqrt{3}}{4}+2\times10^{-4}\leq B\leq \sqrt{\frac{\sqrt{3}-1}{2}} \cdot \frac{\Gamma(\frac{1}{3})\Gamma(\frac{11}{12})}{\Gamma(\frac{1}{4})},

where Γ is the Gamma function. The lower bound was proved by Chen and Gauthier, and the upper bound dates back to Ahlfors and Grunsky. They also gave an upper bound for the Landau constant.

In their paper, Ahlfors and Grunsky conjectured that their upper bounds are actually the true values of B and L.

References

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.