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 \(z=1\)).

The main result reads:

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 \(H\).

The main result reads:

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 \(H\).

Reviewer: Sergei V. Rogosin (Minsk)

##### MSC:

30C55 | General theory of univalent and multivalent functions of one complex variable |

30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |

30C35 | General theory of conformal mappings |

30C75 | Extremal problems for conformal and quasiconformal mappings, other methods |