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The upper and lower bounds on Korenblum’s constant

Wettergren, Åke LU (2020) In Masters' Theses in Mathematical Sciences FMAM05 20201
Mathematics (Faculty of Engineering)
Abstract
The purpose of this thesis is to study complex analysis, the Bergman space and Korenblum's conjecture. This is done in three parts. In the first part the proofs that the conjecture is true are studied, giving lower bounds of Korenblum's constant. The first proof is explained in detail, to make it as accessible as possible to more people. The main differences for a couple of later proofs that improved the lower bound are presented briefly. In the second part the counter examples to the conjecture for larger radii are presented. The first counter examples are explained briefly. The most recent, with the lowest known upper bound of Korenblum's constant, is presented in great detail. In the third part a couple of attempts of improving the... (More)
The purpose of this thesis is to study complex analysis, the Bergman space and Korenblum's conjecture. This is done in three parts. In the first part the proofs that the conjecture is true are studied, giving lower bounds of Korenblum's constant. The first proof is explained in detail, to make it as accessible as possible to more people. The main differences for a couple of later proofs that improved the lower bound are presented briefly. In the second part the counter examples to the conjecture for larger radii are presented. The first counter examples are explained briefly. The most recent, with the lowest known upper bound of Korenblum's constant, is presented in great detail. In the third part a couple of attempts of improving the upper bound are discussed. In the first attempt Blaschke products are used, to be able to place zeros of functions anywhere in the unit disc. In the second attempt the upper bound is analyzed as a variational problem. An optimization algorithm is written to find counter examples for as low radii as possible. The algorithm finds counter examples that are close to the best known, but nothing that is better than what already exists. (Less)
Please use this url to cite or link to this publication:
author
Wettergren, Åke LU
supervisor
organization
alternative title
Den övre och undre begränsningen av Korenblums konstant
course
FMAM05 20201
year
type
H2 - Master's Degree (Two Years)
subject
keywords
Complex Analysis, Bergman Space, Korenblum's constant, Optimization
publication/series
Masters' Theses in Mathematical Sciences
report number
LUTFMA-3421-2020
ISSN
1404-6342
other publication id
2020:E41
language
English
id
9022204
date added to LUP
2020-06-29 14:21:40
date last changed
2020-06-29 14:21:40
@misc{9022204,
  abstract     = {{The purpose of this thesis is to study complex analysis, the Bergman space and Korenblum's conjecture. This is done in three parts. In the first part the proofs that the conjecture is true are studied, giving lower bounds of Korenblum's constant. The first proof is explained in detail, to make it as accessible as possible to more people. The main differences for a couple of later proofs that improved the lower bound are presented briefly. In the second part the counter examples to the conjecture for larger radii are presented. The first counter examples are explained briefly. The most recent, with the lowest known upper bound of Korenblum's constant, is presented in great detail. In the third part a couple of attempts of improving the upper bound are discussed. In the first attempt Blaschke products are used, to be able to place zeros of functions anywhere in the unit disc. In the second attempt the upper bound is analyzed as a variational problem. An optimization algorithm is written to find counter examples for as low radii as possible. The algorithm finds counter examples that are close to the best known, but nothing that is better than what already exists.}},
  author       = {{Wettergren, Åke}},
  issn         = {{1404-6342}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Masters' Theses in Mathematical Sciences}},
  title        = {{The upper and lower bounds on Korenblum’s constant}},
  year         = {{2020}},
}