On separation of singularities of meromorphic functions

V. I. Danchenko

Sbornik. Mathematics, Tome 53 (1986) no. 1, pp. 183-201

Résumé

Let $E$ be an arbitrary bounded proper continuum on $\overline{\mathbf C}$, $\lambda$ a finite collection of pairwise distinct domains that are components of $\overline{\mathbf C}\setminus E$, $f$ a function meromorphic in each domain $G\in\lambda$ and continuous in some neighborhood of $E$, $f_\lambda$ the sum of the principal parts of the Laurent expansions of $f$ with respect to its poles in the union of the domains in $\lambda$, and $n_\lambda$ the degree of the rational function $f_\lambda$. If all the domains $G\in\lambda$ are bounded, then $\|f_\lambda\|_{C(E)}\leqslant\mathrm{const}\cdot n_\lambda\|f\|_{C(E)}$. If $E$ is a rectifiable curve $\Gamma$, then the total variation $\operatorname{Var}(f_\lambda,\Gamma)=\int_\Gamma|f_\lambda'(\zeta)|\cdot|d\zeta|$ of $f_\lambda$ along $\Gamma$ satisfies $\operatorname{Var}(f_\lambda,\Gamma)\leqslant\mathrm{const}\cdot n_\lambda\ln^3(en_\lambda)\|f\|_{C(\Gamma)}V(\Gamma)$, where $V(\Gamma)$ is the supremum of the set $\{\operatorname{Var}(r,\Gamma)\}$ of total variations along $\Gamma$ of all the partial fractions $r(z)=a/(bz+c)$ with $\|r\|_{C(\Gamma)}=1$. Bibliography: 11 titles.