Geodesic
The structure of integrals of the second Loewner--Kufarev differential equation in a particular case
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 55 (2018), pp. 12-21

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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.
Keywords: geometric theory of functions of a complex variable, Loewner–Kufarev differential equation. 
@article{VTGU_2018_55_a1,
     author = {O. V. Zadorozhnaya and V. K. Kochetkov},
     title = {The structure of integrals of the second {Loewner--Kufarev} differential equation in a particular case},
     journal = {Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika},
     pages = {12--21},
     publisher = {mathdoc},
     number = {55},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTGU_2018_55_a1/}
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O. V. Zadorozhnaya; V. K. Kochetkov. The structure of integrals of the second Loewner--Kufarev differential equation in a particular case. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 55 (2018), pp. 12-21. http://geodesic.mathdoc.fr/item/VTGU_2018_55_a1/