Geodesic
On separation of singularities of meromorphic functions
Sbornik. Mathematics, Tome 53 (1986) no. 1, pp. 183-201 
V. I. Danchenko. On separation of singularities of meromorphic functions. Sbornik. Mathematics, Tome 53 (1986) no. 1, pp. 183-201. http://geodesic.mathdoc.fr/item/SM_1986_53_1_a9/
@article{SM_1986_53_1_a9,
     author = {V. I. Danchenko},
     title = {On separation of singularities of meromorphic functions},
     journal = {Sbornik. Mathematics},
     pages = {183--201},
     year = {1986},
     volume = {53},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1986_53_1_a9/}
}
TY  - JOUR
AU  - V. I. Danchenko
TI  - On separation of singularities of meromorphic functions
JO  - Sbornik. Mathematics
PY  - 1986
SP  - 183
EP  - 201
VL  - 53
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1986_53_1_a9/
LA  - en
ID  - SM_1986_53_1_a9
ER  - 
%0 Journal Article
%A V. I. Danchenko
%T On separation of singularities of meromorphic functions
%J Sbornik. Mathematics
%D 1986
%P 183-201
%V 53
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1986_53_1_a9/
%G en
%F SM_1986_53_1_a9

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Gonchar A. A., Grigoryan L. D., “Ob otsenkakh normy golomorfnoi sostavlyayuschei meromorfnoi funktsii”, Matem. sb., 99(141) (1976), 634–638 | Zbl

[2] Grigoryan L. D., “Otsenki normy golomorfnykh sostavlyayuschikh meromorfnykh funktsii v oblastyakh s gladkoi granitsei”, Matem. sb., 100(142) (1976), 156–164. | Zbl

[3] Dolzhenko E. P., “Nekotorye tochnye integralnye otsenki proizvodnykh ratsionalnykh i algebraicheskikh funktsii. Prilozheniya”, Anal. Math., 4:4 (1978), 247–268 | DOI | MR

[4] Dolzhenko E. P., “Ratsionalnye approksimatsii i granichnye svoistva analiticheskikh funktsii”, Matem. sb., 69(111) (1966), 497–524. | Zbl

[5] Sevastyanov E. A., “Ratsionalnaya approksimatsiya i absolyutnaya skhodimost ryadov Fure”, Matem. sb., 107(149) (1978), 227–244 | MR | Zbl

[6] Pekarskii A. A., “Otsenki proizvodnoi integratsii tipa Koshi s meromorfnoi plotnostyu i ikh prilozheniya”, Matem. zametki, 31:3 (1982), 389–402 | MR | Zbl

[7] Andrievskii V. V., “Ob integralnykh otsenkakh proizvodnykh ratsionalnykh funktsii”, Anal. Math., 9:1 (1983), 3–7 | DOI | MR | Zbl

[8] Privalov I. I., Vvedenie v teoriyu funktsii kompleksnogo peremennogo, Nauka, M., 1977 | MR

[9] Danchenko V. I., Otsenki proizvodnykh ratsionalnykh funktsii na kontinuumakh;, Rukopis dep. v VINITI 26.11.82, No 5886-82 Dep., Vladim. politekhn. in-t, Vladimir, 13 pp.

[10] Danchenko V. I., Otsenki variatsii ratsionalnykh funktsii na spryamlyaemykh krivykh, Rukopis dep. v VINITI 08.08.80, No 3515-80 Dep., Vladim. politekhn. in-t, Vladimir, 21 pp.

[11] Danchenko V. I., Nekotorye integralnye i lokalnye otsenki modulei proizvodnykh ratsionalnykh funktsii, Rukopis dep. v VINITI 06.10.82, No 5098-82 Dep., Vladim. politekhn. in-t, Vladimir, 26 pp.