Geodesic
On Holomorphic Motions of~$n$-Symmetric Functions
Matematičeskie zametki, Tome 87 (2010) no. 6, pp. 848-854

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We generalize a problem examined by Duren on the univalence of a family of $n$-symmetric functions generated by integrals of functions of the form $\exp(\lambda \zeta^n)$. Our approach is based on the use of the inverse Faber transform, of the Martio–Sarvas univalence criterion, and of the $\lambda$-lemma of Mañé, Sad, and Sullivan. We also put forward a conjecture on the univalence of a family of $n$-symmetric functions, which is a weakened form of the Danikas–Ruscheweyh conjecture on the univalence of an integral transform of holomorphic functions.
Keywords: $n$-symmetric function, domain with quasiconformal boundary, Danikas–Ruscheweyh conjecture, holomorphic function.
Mots-clés : inverse Faber transform 
@article{MZM_2010_87_6_a5,
     author = {I. R. Kayumov},
     title = {On {Holomorphic} {Motions} of~$n${-Symmetric} {Functions}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {848--854},
     publisher = {mathdoc},
     volume = {87},
     number = {6},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2010_87_6_a5/}
}
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I. R. Kayumov. On Holomorphic Motions of~$n$-Symmetric Functions. Matematičeskie zametki, Tome 87 (2010) no. 6, pp. 848-854. http://geodesic.mathdoc.fr/item/MZM_2010_87_6_a5/