## Free poles of quadratic differentials and non-overlapping domains.(Russian)Zbl 0727.30019

Questions of analysis and approximation, Collect. Sci. Works, Kiev, 23-26 (1989).
[For the entire collection see Zbl 0685.00003.]
Let $${\mathfrak M}_ n$$ be the class of n-tuples $$F_ n=(f_ 1,...,f_ n)$$ of functions which are meromorphic and one-to-one in the unit disc D fulfilling the following properties:
(i) $$E_ i\cap E_ j=\emptyset$$ for $$i\neq j$$, $$i,j=1,2,...,n$$, where $$E_ k=f_ k(D).$$
(ii) $$f_ k(0)=a_ k$$, $$k=1,2,...,n$$, where $$a_ k\in E$$ are given and fulfill the condition (1/n)$$\sum^{n}_{k=1}| a_ k| =1.$$
The following extremal problem in the class $${\mathfrak M}_ n$$ is studied: find $$I_ n=\max_{F_ n\in {\mathfrak M}_ n}\prod^{n}_{k=1}| f'_ k(0)|$$. Theorem 1. For every extremal n-tuple $$\psi_ n=(\psi_ 1,\psi_ 2,...,\psi_ n)\in {\mathfrak M}_ n$$ the set $${\mathbb{C}}\cup \{\infty \}\setminus (\cup^{n}_{k=1}E_ k)$$, where $$E_ k=\psi_ k(D)$$, is composed from a finite number of critical points and trajectories of the quadratic differential $Q(w)dw^ 2=- \sum^{n}_{k=1}(\frac{1}{(a_ k-w)^ 2}+\frac{| a_ k|}{w(a_ k-w)}dw^ 2\text{ and } (\cup^{n}_{k=1}E_ k)\cup \{\infty \}={\bar {\mathbb{C}}}.$ Theorem 2. For $$n=3$$, $$I_ 3=32/3\sqrt{3}$$. The extremal 3-tuple $$\psi_ 3$$ is described explicitely.
Reviewer: J.Fuka

### MSC:

 30C70 Extremal problems for conformal and quasiconformal mappings, variational methods