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/