# zbMATH — the first resource for mathematics

On separation of singularities of meromorphic functions. (English. Russian original) Zbl 0611.30032
Math. USSR, Sb. 53, 183-201 (1986); translation from Mat. Sb., Nov. Ser. 125(167), No. 2, 181-198 (1984).
Let E be an arbitrary proper continuum in $${\bar {\mathbb{C}}}$$, $$\lambda =\{D_ k\}^ a$$finite collection of bounded pairwise distinct connected components of $${\bar {\mathbb{C}}}\setminus E$$, f a function meromorphic in each domain $$D_ k$$ and continuous in some neighbourhood of E, $$F_{\lambda}$$ be the sum of principal parts of the Laurent expansions of f with respect to the poles in $$\cup D_ k$$ and $$n_{\lambda}$$ the degree of the rational function $$F_{\lambda}$$. The main result of the paper is the inequality $$\| F_{\lambda}\| \leq const\cdot n_{\lambda}\| f\|$$ where $$\| \cdot \|$$ is the supremum norm on E. Other interesting results concern the case when E is a rectifiable curve.
Reviewer: A.E.Eremenko

##### MSC:
 30E10 Approximation in the complex plane 30A10 Inequalities in the complex plane
##### Keywords:
rational approximation; Cauchy kernel; Green capacity
Full Text: