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Identifier

000451321

Title

Ο νόμος του λογάριθμου για συναρτήσεις Bloch

Alternative Title

The law of logarithm for block function

Author

Ροδίτου, Χρυσαφίνα

Thesis advisor

Παπαδημητράκης, Μιχάλης

Reviewer

Μήτσης, Θ.
Κολουντζάκης, Μ.

Abstract

The purpose of this thesis is to study some properties regarding a class of functions named Bloch. The main result is the “Law of Iterated Logarithm for Bloch functions” that gives us information about the growth of these functions. The basic idea goes back to Andre Bloch, who introduced a class of functions, which form the so-called Bloch space. During the period from 1925 through 1968 Bloch’s results motivated many mathematicians and the research was extended. From 1969 to the present, a modern approach to the study of these functions has prevailed, using methods of Functional Analysis and Measure Theory. The conformal maps from D onto D will also play an important role in our study. The main theorem, the “Law of Iterated Logarithm”, was initially proved by Nikolai G. Makarov (1990), but we will present it in the form that has been proven by Christian Pommerenke. The thesis is divided into two main sections. In the first section, we will define the Bloch functions, which are functions g analytic in the unit disc such that ||g||B = sup z∈D (1 − |z|2)|g′(z)| < ∞ and we will prove some properties, some of which will be used in the proof of the main theorem. In particular, we will prove that the set of Bloch functions B form a complex Banach space with a suitable norm, that the usual Bloch-seminorm ||g||B = sup z∈D (1 − |z|2)|g′(z)|, g ∈ B is conformally invariant, that H∞ & B and also that if f maps D conformally into C then || log(f − a)||B ≤ 4, a /∈ f(D) and || log f′||B ≤ 6. In the second section, we will formulate and prove the Law of Iterated Logarithm and a corollary of it.

Language

Greek, English

Subject

Νόμος επαναλαμβανόμενου λογάριθμου

Issue date

2022-07-22

Collection

School/Department--School of Sciences and Engineering--Department of Mathematics and Applied Mathematics--Post-graduate theses

Type of Work--Post-graduate theses

Views

260

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