The Growth and Distortion Theorems for Slice Regular Functions
The Growth and Distortion Theorems for Slice Regular Functions
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发表时间：
2014-10
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通讯作者：
G. Ren;X. Wang
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作者：
G. Ren;X. Wang
建立了单位圆盘上单叶全纯函数的切片正则扩张到四元数或Clifford代数集合的尖锐增长和变形定理。结果表明，单叶全纯函数的切片正则扩张到高维的几何函数理论在没有额外的几何假设的情况下成立，这与几个复变量的设置相反，其中增长和扭曲定理一般是不成立的，并且只适用于具有星形或凸假设的某些子类。然而，相应的覆盖定理尚未确定，这表明推广是间接的。
The sharp growth and distortion theorems are established for the slice regular extension to the setting of quaternions or Clifford algebras for a univalent holomorphic function of one complex variable on the unit disc. It turns out that the geometric function theory for the slice regular extension to higher dimensions of a univalent holomorphic function holds without extra geometric assumptions, in contrast to the setting of several complex variables in which the growth and distortion theorems are failed in general and only hold for some subclasses with the starlike or convex assumption. However, the corresponding covering theorem is unsettled, which indicates that the extensions are indirect.