The structure of integrals of the second Loewner–Kufarev differential equation in a particular case

In the geometric theory of functions of a complex variable, the first and the second Loewner–Kufarev differential equations are well known. Considering the first one of them, I. E. Bazilevich pointed out the class of univalent functions in a unit circle, now known as I. E. Bazilevich's class. This paper shows that I. E. Bazilevich's formula can be derived by considering the second Loewner-Kufarev equation with a linear right-hand side. We have also studied a differential equation with a nonlinear right-hand side, rational in a particular case. The problem point in the latter case is to specify a parametric family of regular functions with a positive real part in the unit circle at each fixed value of the parameter. The two lemmas proved in the paper simplify the problem of constructing a right-hand side with a positive real part when considering nonlinear right-hand sides.