Roper-Suffridge extension operator and the lower bound for the distortion
Roper-Suffridge extension operator and the lower bound for the distortion
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DOI：
10.1016/j.jmaa.2004.06.052
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发表时间：
2004-12
影响因子：
1.3
通讯作者：
H. Hamada;G. Kohr
中科院分区：
文献类型：
--
作者：
H. Hamada;G. Kohr
Liczberski-Starkov 在原点附近给出了 ‖DΦn(f)(z)‖ 的尖锐下界，其中 Φ 是 Roper-Suffridge 扩展算子，f 是 C 中单位圆盘上的归一化凸映射。他们给出了一个猜想，即该尖锐下界在欧几里得单位球 Bnin Cn 上成立。在本文中，我们将为更一般的可拓算子和归一化单价映射 f 或归一化凸映射 f 给出 Bn 的尖锐下界。我们将给出线性不变族中映射 f 的下界。我们还将在 Cn 中的有界凸完全莱因哈特域上给出类似的尖锐下界。
Liczberski–Starkov gave a sharp lower bound for ‖DΦn(f)(z)‖ near the origin, where Φnis the Roper–Suffridge extension operator and f is a normalized convex mapping on the unit disk in C. They gave a conjecture that the sharp lower bound holds on the Euclidean unit ball Bnin Cn. In this paper, we will give a sharp lower bound on Bnfor a more general extension operator and for normalized univalent mappings f or normalized convex mappings f. We will give a lower bound for mappings f in a linear invariant family. We will also give a similar sharp lower bound on bounded convex complete Reinhardt domains in Cn.