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Bieberbach's Theorem, Normal Families and Approximation by Single-Slit Maps

Runyeon Odeberg, Robin LU (2022) In Bachelor's Theses in Mathematical Sciences MATK11 20221
Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
Abstract
The main purpose of this thesis is to prove the Bieberbach inequality using
Grönwall’s area theorem, and discuss some consequences of the Montel’s theorem
and Carathéodroty convergence theorem regarding single-slit maps, which is essential
to the proof of de Branges’ theorem.
Popular Abstract
Initially conjectured by Ludwig Bieberbach in 1916, de Branges’ theorem has a
delicate yet surprising statement. It investigates univalent functions, i.e. injective
analytic functions, defined on the unit disc and normalised by
$$f(0) = 0 \quad \text{and} f′(0) = 1.$$
With some entry-level knowledge for analytic functions, we know that such a function
has a unique power series expansion
$$f(z) = z + a_2z^2 + a_3z^3 + \cdots.$
Bieberbach conjectured that $|a_n| \leq n$ for all coefficients an for such functions f and
settled the case for a_2. Others had also proved the statement for special functions
and specific coefficients, but eventually it took mathematicians nearly 70 years to
land a full proof.
Along the endeavours, other... (More)
Initially conjectured by Ludwig Bieberbach in 1916, de Branges’ theorem has a
delicate yet surprising statement. It investigates univalent functions, i.e. injective
analytic functions, defined on the unit disc and normalised by
$$f(0) = 0 \quad \text{and} f′(0) = 1.$$
With some entry-level knowledge for analytic functions, we know that such a function
has a unique power series expansion
$$f(z) = z + a_2z^2 + a_3z^3 + \cdots.$
Bieberbach conjectured that $|a_n| \leq n$ for all coefficients an for such functions f and
settled the case for a_2. Others had also proved the statement for special functions
and specific coefficients, but eventually it took mathematicians nearly 70 years to
land a full proof.
Along the endeavours, other theories have been advanced originally to solve the
Bieberbach conjecture, but later on developed with their own merits. The first
breakthrough of the conjecture was made by Charles Loewner in 1923, who proved
for a3 using Loewner chains and the Loewner differential equation, applied to slit
maps. His theories were extended to the Schramm-Loewner evolution in probability
theory by the end of the 20th century and a Fields Medal was awarded for some
related research. (Less)
Please use this url to cite or link to this publication:
author
Runyeon Odeberg, Robin LU
supervisor
organization
course
MATK11 20221
year
type
M2 - Bachelor Degree
subject
keywords
complex analysis, Bieberbach's theorem, Bieberbach inequality, de Branges' theorem, single-slit maps, univalent functions, normal families
publication/series
Bachelor's Theses in Mathematical Sciences
report number
LUNFMA-4133-2022
ISSN
1654-6229
other publication id
2022:K6
language
English
id
9090412
date added to LUP
2023-08-28 18:10:59
date last changed
2023-08-28 18:10:59
@misc{9090412,
  abstract     = {{The main purpose of this thesis is to prove the Bieberbach inequality using
Grönwall’s area theorem, and discuss some consequences of the Montel’s theorem
and Carathéodroty convergence theorem regarding single-slit maps, which is essential
to the proof of de Branges’ theorem.}},
  author       = {{Runyeon Odeberg, Robin}},
  issn         = {{1654-6229}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Bachelor's Theses in Mathematical Sciences}},
  title        = {{Bieberbach's Theorem, Normal Families and Approximation by Single-Slit Maps}},
  year         = {{2022}},
}