The sharp estimates of all homogeneous expansions for a class of quasi-convex mappings on the unit polydisk in ℂ n

Abstract

In this paper, the sharp estimates of all homogeneous expansions for f are established, where f(z) = (f1(z), f2(z), …, fn(z))′ is a k-fold symmetric quasi-convex mapping defined on the unit polydisk in ℂn and $$ \begin{gathered} \frac{{D^{tk + 1} + f_p \left( 0 \right)\left( {z^{tk + 1} } \right)}} {{\left( {tk + 1} \right)!}} = \sum\limits_{l_1 ,l_2 ,...,l_{tk + 1} = 1}^n {\left| {apl_1 l_2 ...l_{tk + 1} } \right|e^{i\tfrac{{\theta pl_1 + \theta pl_2 + ... + \theta pl_{tk + 1} }} {{tk + 1}}} zl_1 zl_2 ...zl_{tk + 1} ,} \hfill \\ p = 1,2,...,n. \hfill \\ \end{gathered} $$ Here i = \( \sqrt { - 1} \), θplq ∈ (−θ,θ] (q = 1, 2, …, tk + 1), l1, l2, …, ltk+1 = 1, 2, …, n, t = 1, 2, …. 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.