Some extremal problems for multiply connected domains. (Russian) Zbl 0727.30020

Bakhtin, A. K.

Questions of analysis and approximation, Collect. Sci. Works, Kiev, 19-22 (1989).

[For the entire collection see Zbl 0685.00003.]
Let \(E_ n\subset {\bar {\mathbb{C}}}={\mathbb{C}}\cup \{\infty \}\) by an n-times connected region containing \(z=\infty\) and bounded by n circles. Let denote \(\Sigma_ n\) the class of functions meromorphic in \(E_ n\), which are one-to-one in \(E_ n\) and of the form \(f(z)=\epsilon z+\alpha_ 0+\alpha_ 1/z\), \(| \epsilon | =d\), in the neighborhood of \(z=\infty\). Let \({\bar {\mathbb{C}}}\setminus f(E_ n)=\cup^{n}_{i=1}K_ i\), where \(K_ i\) are mutually disjoint continua. Let denote \[ d_ n(f)=\max_{c_ i\in K_ i}[\prod_{1\leq i<j\leq n}| c_ i-c_ j|^{2/(n(n-1))}]. \] The following extremal problem is studied: to find \(\max_{f\in \Sigma_ n}d_ n(f).\)
Theorem. The compact solving the extremal problem is composed from n mutually disjoint continua \(K_ i\), \(i=1,2,...,n\), each of them being the closure of a finite number of arcs of trajectories of the quadratic differential \[ Q(w)dw^ 2=-\sum \frac{dw^ 2}{(c^ 0_ i-w)(c^ 0_ j-w)}. \] The case \(n=3\) is discussed in detail.

Reviewer: J.Fuka

MSC:

30C70

Extremal problems for conformal and quasiconformal mappings, variational methods

Keywords:

trajectories; quadratic differential

Citations:

Zbl 0685.00003