Geodesic
On the Univalence of Derivatives of Functions which are Univalent in Angular Domains
Matematičeskie zametki, Tome 82 (2007) no. 6, pp. 885-890.

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

We consider functions $f$ that are univalent in a plane angular domain of angle $\alpha\pi$, $0\alpha\le2$. It is proved that there exists a natural number $k$ depending only on $\alpha$ such that the $k$th derivatives $f^{(k)}$ of these functions cannot be univalent in this angle. We find the least of the possible values of for $k$. As a consequence, we obtain an answer to the question posed by Kiryatskii: if $f$ is univalent in the half-plane, then its fourth derivative cannot be univalent in this half-plane.
Keywords: univalent function, holomorphic function, Bieberbach's conjecture, Koebe function, Weierstrass theorem. 
@article{MZM_2007_82_6_a7,
     author = {S. R. Nasyrov},
     title = {On the {Univalence} of {Derivatives} of {Functions} which are {Univalent} in {Angular} {Domains}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {885--890},
     publisher = {mathdoc},
     volume = {82},
     number = {6},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2007_82_6_a7/}
}
TY  - JOUR
AU  - S. R. Nasyrov
TI  - On the Univalence of Derivatives of Functions which are Univalent in Angular Domains
JO  - Matematičeskie zametki
PY  - 2007
SP  - 885
EP  - 890
VL  - 82
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2007_82_6_a7/
LA  - ru
ID  - MZM_2007_82_6_a7
ER  - 
%0 Journal Article
%A S. R. Nasyrov
%T On the Univalence of Derivatives of Functions which are Univalent in Angular Domains
%J Matematičeskie zametki
%D 2007
%P 885-890
%V 82
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2007_82_6_a7/
%G ru
%F MZM_2007_82_6_a7
S. R. Nasyrov. On the Univalence of Derivatives of Functions which are Univalent in Angular Domains. Matematičeskie zametki, Tome 82 (2007) no. 6, pp. 885-890. http://geodesic.mathdoc.fr/item/MZM_2007_82_6_a7/

[1] S. M. Shah, S. Y. Trimble, “Univalent functions with univalent derivatives”, Bull. Amer. Math. Soc., 75 (1969), 153–157 | DOI | MR | Zbl

[2] S. M. Shah, S. Y. Trimble, “Univalent functions with univalent derivatives. II”, Trans. Amer. Math. Soc., 144 (1969), 313–320 | DOI | MR | Zbl

[3] S. M. Shah, S. Y. Trimble, “Univalent functions with univalent derivatives. III”, J. Math. Mech., 19 (1969), 451–460 | MR | Zbl

[4] J. Kirjackis, “On the existence of functions being univalent in half-plane together with their derivatives”, Nonlinear Anal. Model. Control, 6:2 (2001), 43–50 | MR | Zbl

[5] I. A. Aleksandrov, V. V. Sobolev, “Ekstremalnye zadachi dlya nekotorykh klassov funktsii, odnolistnykh v poluploskosti”, Ukr. matem. zhurn., 22:3 (1970), 291–307 | MR | Zbl

[6] F. G. Avkhadiev, “O nekotorykh odnolistnykh otobrazheniyakh poluploskosti”, Trudy seminara po kraevym zadacham, 11 (1974), 3–8 | MR

[7] G. J. Dimkov, J. Stankiewicz, Z. Stankiewicz, “On a class of starlike functions defined in a halfplane”, Ann. Polon. Math., 55 (1991), 81–86 | MR | Zbl

[8] A. M. Zakharov, D. V. Prokhorov, “Mnozhestvo znachenii funktsii i ee proizvodnoi v klasse odnolistnykh otobrazhenii poluploskosti”, Izv. vuzov. Ser. matem., 1993, no. 2, 33–37 | MR | Zbl

[9] S. E. Demin, “Izoperimetricheskaya zadacha iskazheniya dlya odnolistnykh funktsii Montelya”, Sib. matem. zhurn., 37:1 (1996), 108–116 | MR | Zbl

[10] V. V. Goryainov, I. Ba, “Polugruppa konformnykh otobrazhenii verkhnei poluploskosti v sebya s gidrodinamicheskoi normirovkoi na beskonechnosti”, Ukr. matem. zhurn., 44:10 (1992), 1320–1329 | MR | Zbl

[11] A. Lecko, “On the class of functions defined in a halfplane and starlike with respect to a boundary point”, Ann. Polon. Math., 79:1 (2002), 67–83 | DOI | MR | Zbl

[12] N. N. Pascu, N. R. Pascu, “Convex functions in a half-plane”, paper No 102, JIPAM. J. Inequal. Pure Appl. Math., 4:5 (2003) | MR | Zbl

[13] G. M. Goluzin, Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Nauka, M., 1966 | MR | Zbl

[14] L. de Brange, “A proof of the Bieberbach conjecture”, Acta Math., 154:1–2 (1985), 137–152 | DOI | MR | Zbl