The upper and lower bounds on Korenblum’s constant

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.