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Abstract:
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In this dissertation we consider various versions of the Schwarz lemma and theSchwarz-Pick lemma for holomorphic, harmonic and harmonic quasiregular mappings. Inaddition, in order to present new results, an overview of the results that can be considered asclassical is given. As one of the most important consequence of the Schwarz-Pick lemma forholomorphic mappings, an introduction of the hyperbolic distancedΩon the simply connecteddomainsΩ C(such thatΩ6=C) is given in details, as well as the connection of that distanceand holomorphic mappings. All versions of the Schwarz lemma and the Schwarz-Pick lemma for harmonic mappingsare shown as assertions analogous to the corresponding claims for holomorphic mappings.In the proofs of these assertions, the properties of the hyperbolic distance and Euclideanproperties of hyperbolic disks are used. Firstly, we considered some versions of the Schwarzlemma for harmonic mappings from the unit disk to the interval (-1,1) and then for harmonicmappings of the unit disk into itself, without the assumption thatz= 0is mapped to itself bythe corresponding map. Thereby, the corresponding inequalities were shown to be sharp andextremal mappings were found. By using the strip and half plane method, simple proofs ofthe Schwarz-Pick lemma for real-valued harmonic mappings are given, as well as the simpleproofs of their corollaries that are formulated in terms of corresponding hyperbolic distances.For both holomorphic and harmonic mappings a version of the Schwarz lemma have beenformulated and proved in the case where the values of these mappings and values of thenorms of their differentials, at the pointz= 0, are given. Also, in that case we showed thatthe corresponding inequalities are sharp and extremal mappings were found. It has also beenshown that the same methods can be used to obtain Harnack’s inequalities for harmonicmappings, as well as for their generalizations.Furthermore, we give simple proofs of a version of the Schwarz-Pick lemma for harmonicquasiregular mappings whose codomain is a half plane or a strip. One version of the Schwarzlemma for harmonic quasiregular mappings from the unit disk into a strip is obtained thanksto the appropriate (which seems unexpected) inequality satisfied by the Euclidean and hyper-bolic distances on the strip. By using the properties of the Gaussian curvature we also showthat harmonic quasiconformal mappings of the hyperbolic domain into convex hyperbolicdomain are quasi-isometries of the corresponding metric spaces.The introduction of the hyperbolic distance is shown in two ways. The first way is classi-cal one. Starting from the hyperbolic metric on the unit disk, first we define the hyperboliclength of theC1curve and then the hyperbolic distance between two given points. Thesecond one is based on the axiomatic foundation of the absolute plane geometry. Startingfrom the theorem related to the existence and the uniqueness (up to the unit for length) ofthe distance in the absolute plane (which is in accordance with the basic geometry relations- between and congruence), we simultaneously derive the formula for that distance in twomodels of that plane. One of these models is the set of complex numbersC, observed as amodel of the Euclidean plane and the second one is the unit disk that is considered as thePoincaré disk model of hyperbolic plane. |