Geodesic
On the possibility of division and involution to a fractional power in the algebra of rational functions
Izvestiya. Mathematics, Tome 30 (1988) no. 2, pp. 385-393 
P. V. Paramonov. On the possibility of division and involution to a fractional power in the algebra of rational functions. Izvestiya. Mathematics, Tome 30 (1988) no. 2, pp. 385-393. http://geodesic.mathdoc.fr/item/IM2_1988_30_2_a10/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Suppose that a function $f(z)$ satisfies a Lipschitz condition with an arbitrary positive element on a compact set $X$ in $\mathbf C$ and can be uniformly approximated on $X$ by rational functions. If $q>1$ and some branch of $(f(z))^q$ is continuous on $X$, then this branch can also be approximated on $X$ by rational functions. Also, an example is given of a compact set $X$ and two functions $f(z)$ and $g(z)$ uniformly approximable on $X$ by rational functions and with ratio $g(z)/f(z)$ naturally (uniquely) defined and continuous on $X$ but not approximable by rational functions. Bibliography: 7 titles.

[1] Davie A. M., Oksendal B. K., “Rational approximations on the union of sets”, Proc. Amer. Math. Soc., 29:3 (1971), 581–584 | DOI | MR | Zbl

[2] Paramonov P. V., “O vzaimosvyazi lokalnykh i globalnykh approksimatsii golomorfnymi funktsiyami”, Izv. AN SSSR. Ser. matem., 46:1 (1982), 100–116 | MR | Zbl

[3] Paramonov P. V., “Ob odnom dostatochnom uslovii priblizhaemosti funktsii ratsionalnymi drobyami”, Dokl. AN SSSR, 268:2 (1983), 292–295 | MR | Zbl

[4] Vitushkin A. G., “Analiticheskaya emkost v zadachakh teorii priblizhenii”, Uspekhi matem. nauk, 22:6 (1967), 141–199 | MR

[5] Melnikov M. S., “Metricheskie svoistva analiticheskoi $\alpha$-emkosti i priblizheniya analiticheskikh funktsii s usloviem Geldera ratsionalnymi funktsiyami”, Matem. sb., 79:1 (1969), 118–127 | MR

[6] Gonchar A. A., “O primerakh needinstvennosti analiticheskikh funktsii”, Vestnik Mosk. un-ta. Matem., mekhan., 1964, no. 1, 37–43 | Zbl

[7] Gamelin T. V., Ravnomernye algebry, Mir, M., 1973 | Zbl