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Evolution families of conformal mappings with fixed points and the Löwner-Kufarev equation. (English. Russian original) Zbl 1321.30012
Sb. Math. 206, No. 1, 33-60 (2015); translation from Mat. Sb. 206, No. 1, 39-68 (2015).
The author develops an analogue of the Löwner-Pommerenke parametric method to study interior problems in the theory of univalent functions with constrains on the angular derivative at a boundary point. To do this, he utilizes the class $${\mathcal L}[0,1]$$ of univalent conformal mappings $$f$$ of the unit disc $${\mathbb D}$$ onto itself ($$f(0) = 0$$, and $$f$$ has a finite angular derivative at $$z=1$$).
Theorem. Let $$\{w_{t,s}: 0 \leq s \leq t\leq T\}$$ be an evolution family in $${\mathcal L}[0,1]$$ and let $$\beta(t) = w^{\prime}_{t,0}(1)$$ be an absolutely continuous function on $$[0, T]$$. Then for any $$z\in {\mathbb D}$$ and $$s\in [0, T]$$, the function $$t \mapsto w_{t,s}(z)$$ is absolutely continuous on $$[s, T]$$ and, for almost all $$t$$, $\frac{\partial}{\partial t} w_{t,s}(z) = - \frac{\beta^{\prime}(t)}{\beta(t)} w_{t,s}(z) (1 - w_{t,s}(z)) H(w_{t,s}(z), t),$ where $$H(z,t)$$ is a function defined on $${\mathbb D}\times [0, T]$$, $$z$$-holomorphic and $$t$$-measurable, and for almost all $$t\in [0, T]$$ the function $$H(\cdot,t)$$ is represented in the form $H(z,t) = \int\limits_{\mathbb T} \frac{1 - \chi}{1 - \chi z} d \mu (\chi)$ with some probability measure $$\mu$$ on $${\mathbb T}$$.
Under the conditions of this theorem, by changing the time scale, a normalized evolution family with $$\beta(t) = e^t$$, $$0 \leq t \leq T$$, is obtained.
It is also shown that each mapping $$f$$ in the semigroup $${\mathcal L}[0,1]$$ can be embedded into an evolution family that is generated by some infinitesimal generating function $$H$$.