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


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


Zbl 0685.00003