|
|
| |
| On Univalence of the Power Deformation $z(\frac{f(z)}{z})^c$ |
| |
Citation： |
Yong Chan KIM,Toshiyuki SUGAWA.On Univalence of the Power Deformation $z(\frac{f(z)}{z})^c$[J].Chinese Annals of Mathematics B,2012,33(6):823~830 |
| Page view： 1882
Net amount： 1597 |
Authors： |
Yong Chan KIM; Toshiyuki SUGAWA; |
Foundation： |
Yeungnam University (2011) (No.211A380226) and the JSPS Grant-in-Aid for Scientific Research (B)(No. 22340025). |
|
|
| Abstract： |
The authors mainly concern the set $U_f$ of $c\in\C$ such that the power deformation $z\big(\frac{f(z)}z\big)^c$ is univalent in the unit disk $|z|<1$ for a given analytic univalent function $f(z)=z+a_2z^2+\cdots$ in the unit disk. It is shown that $U_f$ is a compact, polynomially convex subset of the complex plane $\C$ unless $f$ is the identity function. In particular, the interior of $U_f$ is simply connected. This fact enables us to apply various versions of the $\lambda$-lemma for the holomorphic family $z\big(\frac{f(z)}{z}\big)^c$ of injections parametrized over the interior of $U_f.$ The necessary or sufficient conditions for $U_f$ to contain $0$ or $1$ as an interior point are also given. |
Keywords： |
Univalent function, Holomorphic motion, Quasiconformal extension, Grunsky inequality, Univalence criterion |
Classification： |
30C55, 30C62 |
|
|
Download PDF Full-Text
|
