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Abstract:
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In this PhD thesis we investigate bounds of the gradient of harmonic and
harmonic quasiconformal mappings. We also discuss such bounds for functions that are harmonic
with respect to the hyperbolic metric or certain other metrics. This research has
been motivated by some recent results about Lipschitz-continuity of quasiconformal mappings
that satisfy the Laplace gradient inequality. More precisely, the mappings we consider
are solutions of the Dirichlet problem for the Poisson equation and can be considered as a
generalization of harmonic mappings. Besides the ball, we also work with general domains on
which solutions of the Dirichlet problem are defined, as well as general codomains. Finally,
we announce new results that have been formulated for regions of C1,α-smoothness, both as
the domain and the codomain.
Besides presenting the main results, we give an overview of general notions from differential
geometry and recall some of the properties of hyperbolic metric in an n-dimensional ball. We
also state properties of harmonic and sub-harmonic functions with respect to the hyperbolic
metric, which are analogous to some classical results from the theory if harmonic functions
and Hardy’s theory. It turns out that the gradients of hyperbolic harmonic functions behave
differently from those of euclidean harmonic functions. A similar conclusion is obtained
for the family of Tα-harmonic functions. Namely, unlike the space of harmonic functions,
the solution of the Dirichlet problem in the space of Tα-harmonic functions is shown to be
Lipschitz-continuous when so is the boundary function. In addition, we investigate Höldercontinuity
of the solution of the Dirichlet problem for the Poisson equation in the euclidean
and hyperbolic metric.
We will present versions of the Schwarz lemma on the boundary for pluriharmonic mappings
in Hilbert and Banach spaces. These results will follow from the version of the Schwarz
lemma for harmonic mappings from the unit disc to the interval (1, 1) without the assumption
that the point z = 0 maps to itself. Furthermore, we show a version of the boundary
Schwarz lemma for harmonic mappings from a ball to a ball, not necessarily of the same
dimension. The proof uses a version of the Schwarz lemma for multivariable functions, first
considered by Burget. This result is obtained by integrating the Poisson kernel over so-called
polar caps. The assumption that point z = 0 maps to itself is again not needed, thus yielding
a generalization of a recent result by D. Kalaj. At the end of this section, it is demonstrated
that the analogous result is false in the case of hyperbolic harmonic functions. In a certain
sense, this means that the Hopf lemma is not valid for hyperbolic harmonic functions.
Amongst various versions of the Schwarz lemma, we have been investigating bounds of
the modulus for classes of holomorphic functions f on the unit disc whose index If fulfils certain
geometric conditions. These classes are a generalization of the star and α-star functions,
previously investigated by B. N. Örnek. Our method is based on using Jack’s lemma and can
be applied in certain more general cases. As an illustration, we derive the sharp bounds for
the modulus of a holomorphic function f with index If whose codomain is a vertical strip,
as well as bounds for the modulus of the derivative of f at point z = 0. Moreover, we give
a bound for the rate of growth of the modulus of holomorphic functions on disk U that map
point z = 0 to itself and whose codomain is a vertical strip. |