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Identifier |
000451321 |
Title |
Ο νόμος του λογάριθμου για συναρτήσεις Bloch |
Alternative Title |
The law of logarithm for block function |
Author
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Ροδίτου, Χρυσαφίνα
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Thesis advisor
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Παπαδημητράκης, Μιχάλης
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Reviewer
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Μήτσης, Θ.
Κολουντζάκης, Μ.
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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.
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Language |
Greek, English |
Subject |
Νόμος επαναλαμβανόμενου λογάριθμου |
Issue date |
2022-07-22 |
Collection
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School/Department--School of Sciences and Engineering--Department of Mathematics and Applied Mathematics--Post-graduate theses
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Type of Work--Post-graduate theses
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Permanent Link |
https://elocus.lib.uoc.gr//dlib/5/b/d/metadata-dlib-1664347033-582447-6273.tkl
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Views |
299 |