|
|
| |
| Quasi-convex Mappings on the Unit Polydisk in $\mathbb{C}^{n}$ |
| |
Citation： |
Xiaosong LIU,Taishun LIU.Quasi-convex Mappings on the Unit Polydisk in $\mathbb{C}^{n}$[J].Chinese Annals of Mathematics B,2011,32(2):241~252 |
| Page view： 1929
Net amount： 1467 |
Authors： |
Xiaosong LIU; Taishun LIU; |
Foundation： |
the National Natural Science Foundation of China (Nos. 10971063, 11061015), the
Major Program of Zhejiang Provincial Natural Science Foundation of China (No. D7080080) and the
Guangdong Provincial Natural Science Foundation of China (No. 06301315). |
|
|
| Abstract： |
In this paper, the sharp estimates of all homogeneous expansions for
$f$ are established, where $f(z)=(f_1(z),f_2(z),\cdots,f_n(z))'$ is
a $k$-fold symmetric quasi-convex mapping defined on the unit
polydisk in $\Cn$ and
\frac{D^{tk+1}f_p(0)(z^{tk+1})}{(tk+1)
!}=\sum\limits_{l_1,l_2,\cdots,l_{tk+1}=1}^n|a_{p\,l_1l_2\cdots
l_{tk+1}}|
\rme^{\rmi\frac{\theta_{p\,l_1}+\theta_{pl_2}+\cdots+\theta_{pl_{tk+1}}}{tk+1}}z_{l_1}z_{l_2}\cdots
z_{l_{tk+1}},\p=1,2,\cdots,n.
Here $\rmi=\sqrt{-1},\ \theta_{pl_q}\in (-\pi, \pi](q=1,2,\cdots,tk+1),\ l_1,l_2,\cdots,l_{tk+1}=1,2,\cdots,n,t=1,2,\cdots$. Moreover, as corollaries, the sharp upper bounds of
growth theorem and distortion theorem for a $k$-fold symmetric
quasi-convex mapping are established as well. These results show
that in the case of quasi-convex mappings, Bieberbach conjecture in
several complex variables is partly proved, and many known results
are generalized. |
Keywords： |
Estimates of all homogeneous expansions, Quasi-convex mapping,
Quasi-convex mapping of type A, Quasi-convex mapping of type B |
Classification： |
32A30, 32H02 |
|
|
Download PDF Full-Text
|
|
|
|
