## Loewner chains in the unit disk.(English)Zbl 1209.30011

A family $$(\varphi_{s,t})_{0\leq s\leq t<\infty}$$ of holomorphic self-maps of the unit disk $$\mathbb D$$ is an evolution family of order $$d\in[1,\infty]$$ if $$\varphi_{s,s}=\text{id}_{\mathbb D}$$, $$\varphi_{s,t}=\varphi_{u,t}\circ\varphi_{s,u}$$ for all $$0\leq s\leq u\leq t<\infty$$, and, for all $$z\in\mathbb D$$ and $$T>0$$, there exists a non-negative function $$k_{z,T}\in L^d\big([0,T],\mathbb R\big)$$ such that
$\big|\varphi_{s,u}(z)-\varphi_{s,t}(z)\big|\leq\int_u^tk_{z,T}(\xi)d\xi, \qquad 0\leq s\leq u\leq t\leq T.$
Correspondingly, a family $$(f_t)_{0\leq t<\infty}$$ of holomorphic maps of $$\mathbb D$$ is a Loewner chain of order $$d$$ if the $$f_t$$ are univalent, $$f_s(\mathbb D)\subset f_t(\mathbb D)$$ for all $$0\leq s<t<\infty$$, and, for any compact set $$K\subset\mathbb D$$ and all $$T>0$$, there exists a non-negative function $$k_{K,T}\in L^d\big([0,T],\mathbb R\big)$$ such that
$\big|f_s(z)-f_t(z)\big|\leq\int_s^tk_{K,T}(\xi)d\xi, \qquad z\in K,\; 0\leq s\leq t\leq T.$
The main results of the paper concerning relations between Loewner chains and evolution families are stated in the following theorems.
Theorem 1.3: For any Loewner chain $$(f_t)$$ of order $$d$$, let $$\varphi_{s,t}:=f_t^{-1}\circ f_s$$, $$0\leq s\leq t$$. Then $$(\varphi_{s,t})$$ is an evolution family of the same order $$d$$. Conversely, for any evolution family $$(\varphi_{s,t})$$, there exists a Loewner chain $$(f_t)$$ of the same order $$d$$ such that $$f_t\circ\varphi_{s,t}=f_s$$, $$0\leq s\leq t$$.
A Loewner chain $$(f_t)$$ is normalized if $$f_0(0)=0$$ and $$f_0'(0)=1$$.
Theorem 1.6: Let $$(\varphi_{s,t})$$ be an evolution family. Then there exists a unique normalized Loewner chain $$(f_t)$$ associated (in the sense of Theorem 1.3) with $$(\varphi_{s,t})$$ such that $$\bigcup_{t\geq0}f_t(\mathbb D)$$ is either an Euclidean disk or the whole complex plane $$\mathbb C$$. Moreover, the following statements are equivalent:
(i) the family $$(f_t)$$ is the only normalized Loewner chain associated with the evolution family $$(\varphi_{s,t})$$;
(ii) for all $$z\in\mathbb D$$,
$\beta(z):=\lim_{t\to\infty}\frac{|\varphi'_{0,t}(z)|}{1-|\varphi_{0,t}(z)|^2}=0;$
(iii) there exist at least one point $$z\in\mathbb D$$ such that $$\beta(z)=0$$;
(iv) $$\bigcup_{t\geq0}f_t(\mathbb D)=\mathbb C$$.
Theorem 1.7: Suppose that under conditions of Theorem 1.6, $$\Omega:=\bigcup_{t\geq 0}f_t(\mathbb D)\neq\mathbb C$$. Then $$\Omega=\{z: |z|<1/\beta(0)\}$$, and the set $$\mathcal L[(\varphi_{s,t})]$$ of all normalized Loewner chains $$(g_t)$$ associated with the evolution family $$(\varphi_{s,t})$$ is given by the formula
$\mathcal L\big[(\varphi_{s,t})\big]=\bigg\{(g_t)_{t\geq0}: g(z)=\frac{h\big(\beta(0)f_t(z)\big)}{\beta(0)},\quad h\in S\bigg\}.$
Here $$S$$ is the class of all univalent holomorphic functions $$h$$ in $$\mathbb D$$, $$h(0)=0$$, $$h'(0)=1$$.
Besides that, the authors find an analogue of the Loewner-Kufarev partial differential equation and show that there is a one-to-one correspondence between the concept of generalized Loewner chains and the generalized Berkson-Porta vector fields. They consider the special case of evolution families induced by semigroups of holomorphic functions in $$\mathbb D$$ and show that the uniqueness of the Kœnigs function is a consequence of Theorems 1.3 and 1.6.

### MSC:

 30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination 34M15 Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable

### Keywords:

Loewner chains; evolution families; Herglotz vector fields
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### References:

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