Parametric representation and asymptotic starlikeness in ℂⁿ
Parametric representation and asymptotic starlikeness in ℂⁿ
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ℂⁿ 中的参数表示和渐进星形
DOI：
10.1090/s0002-9939-08-09392-1
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
M. Kohr
中科院分区：
文献类型：
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作者：
Ian D. Graham;H. Hamada;G. Kohr;M. Kohr
本文考虑欧氏空间C”中渐近星形的概念。在最大范数的情况下，渐近星形性由Poreda引入。我们稍微修改了他的定义，增加了一个有界性条件。我们证明了参数表示的概念，其中出现在Loewner理论可以表征方面的渐近星形，即它们是等价的概念。（不需要Poreda的正则性假设。特别地，aE（-Tr/2，71-/2）型星形映射和螺形映射是渐近星形的.因此，这个概念是对星形的自然概括。然而，我们给出了一个例子的螺旋状映射的线性算子是不是渐近星形。在一个复变量的情况下，类S中的任何函数都是渐近星形的;然而，在n 2维中，这不再是真的。
In this paper we consider the notion of asymptotic starlikeness in the Euclidean space C". In the case of the maximum norm, asymptotic starlikeness was introduced by Poreda. We have modified his definition slightly, adding a boundedness condition. We prove that the notion of parametric representation which arises in Loewner theory can be characterized in terms of asymptotic starlikeness; i.e. they are equivalent notions. (A regularity assumption of Poreda is not needed.) In particular, starlike mappings and spirallike mappings of type a £ (-Tr/2,71-/2) are asymptotically starlike. Therefore this notion is a natural generalization of starlikeness. However, we give an example of a spirallike mapping with respect to a linear operator which is not asymptotically starlike. In the case of one complex variable, any function in the class S is asymptotically starlike; however, in dimension n 2 this is no longer true.