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Abstract:
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This thesis has been written under the supervision of my mentor Prof. Miodrag Mateljevi
c, and my co-mentor dr. Vladimir Bo zin at the University of Belgrade in the academic
year 2012-2013. The topic of this thesis is Complex analysis related with geometric
function theory, more precisely the theory of quasiconformal mappings in the Euclidean
n-dimensional space. For good survey of the eld, see F. W. Gehring [20] in the handbook
of K uhnau [33] which also contains many other surveys on quasiconformal mappings
and related topics. The main source in this dissertation is J. V ais al a [67]. The thesis
is divided into three chapters. Chapter 1 is divided into 5 sections. In this chapter,
we focus on quasiconformal mappings in Rn and discuss various equivalent de nitions.
We give The Modulus of family of curves in the rst section, geometric de nition of
quasiconformal space mappings in second section, analytic de nition of quasiconformal
space mappings in third section, equivalence of the de nitions in fourth section, and the
Beltrami equation in fth section. Chapter 2 is divided into 5 sections. We begin by
generalizing the class of Lip (
), 0 < 1, and some properties of that class. Chapter
2 is devoted to understanding the properties by introducing the notion of Linearity,
Di erentiability, and majorants. A majorant function is a certain generalization of the
power functions t , this is done in the rst section. In the second section we introducing
the notion of moduli of continuity with its Some Properties which gotten from I.M.
Kolodiy, F. Hildebrand paper [39]. In third section we produced harmonic mapping as
preliminary for the fourth section which including subharmonicity of jfjq of harmonic
quasiregular mapping in space. In the last section we introducing estimation of the Poisson
kernel which were extracted from Krantz paper [42]. Chapter 3 is divided into 3
sections. This chapter is include the main result in this dissertation. In this chapter we
prove that !u( ) C!f ( ), where u :
! Rn is the harmonic extension of a continuous
map f : @
! Rn, if u is a K-quasiregular map and
is bounded in Rn with C2 boundary.
Here C is a constant depending only on n, !f and K and !h denotes the modulus
of continuity of h. We also prove a version of this result for !-extension domains with
c-uniformly perfect boundary and quasiconformal mappings, and we state some results
regarding HQC self maps of the quadrant Q = fz : z = x + iy; x; y > 0g. |