One and several complex variables of univalent functions

In this thesis univalent functions of one and several complex variables properties were investigated and a subclass of univalent functions was obtained by using integral operatör. The properties of this new class has been examined by applying differential subordination rule. The theoric perspectives of Loewner equation for several variables of univalent functions were also analyzed. Lipschitz regularity was taken into consideration for general Loewner chains and their transition mappings as a result of this, it was showen that an arbitrary Loewner chain corresponds Loewner differential equation and their transition mappings corresponds to a initial value problem. One of the most important differences between Loewner theory in one dimension and in higher dimensions was shown as the role played by parametric representation. In contrast to the situation in one variable, in several variables there exist Loewner chains such that the initial elemant does not have parametric representation. In the existence theorems for the Loewner differential equation in several variables were improved as a consequence of a theorem that the analog of the Caratheodory class in several variables is compact.