Application of the Loewner-Kufarev theory to the construction of a parametric set of univalent functions of a certain form

This work relates to the theory of Loewner-Kufarev differential equations, which are a part of the geometric function theory. We apply the well-known second Loewner-Kufarev differential equation to construct a parametric family of univalent functions in the unit disk $g(z, t)$ for each fixed non-negative value of the parameter $t$ generalizing the known parametric families. The article also uses various alternative approaches and provides their comparative analysis. The results of the study can be considered as one sufficient condition for the uniqueness of regular functions in a unit disk. Leading Russian scientists made a great contribution to the development of the geometric function theory based the variational-parametric method for studying functionals and found some Loewner-Kufarev differential equations.
There are three sections in the work. The first one applies the Loewner-Kufarev equation to construct a parametric set of univalent functions of a certain type. In the second section, we introduce a special class of regular functions in the unit disk with a fixed convex function, and prove the univalence property for functions of this class. Here we also show one more method for constructing a parametric family of univalent functions different from the methods described in the first paragraph. The third section is devoted to alternative methods for constructing one-parameter sets of univalent functions.