SPHERICAL LINEAR INVARIANCE AND UNIFORM LOCAL SPHERICAL CONVEXITY
SPHERICAL LINEAR INVARIANCE AND UNIFORM LOCAL SPHERICAL CONVEXITY
复制标题
球面线性不变性和均匀局部球凸性
DOI：
10.1142/9789814355896_0012
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发表时间：
1992
期刊：
影响因子：
--
通讯作者：
D. Minda
中科院分区：
文献类型：
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作者：
Wancang Ma;D. Minda
本文是两篇较早的论文（[3]，[4]）的继续，因为在这三篇论文中，我们都研究了与线性不变性和一致局部凸性有关的思想，但在这三篇论文中，我们研究的是不同几何中的思想。我们假设读者对这两篇论文有一定的熟悉程度，因为我们有时会省略与这两篇论文中的相应证明非常相似的证明细节;在这种情况下，总是提供对早期论文的明确引用。本文讨论了从单位圆盘D到黎曼球面P的局部施利希特亚纯函数和从复平面C到黎曼球面的局部单叶亚纯函数的球面线性不变性。我们的两个较早的文件处理的线性不变性和一致的局部凸性局部单叶函数从磁盘D的复平面C或单位磁盘。本文将线性不变性的概念推广到局部单叶亚纯函数f：DP或f：C→ P，并将其与欧氏线性不变性的定义（[11]，[3]）相对应，而不是与双曲线性不变性的定义[4]相对应.在欧氏和双曲线性不变性的情况下，施利希特函数的阶小于或等于2;对于球面线性不变性没有类似的结果，因为单位圆盘D中正规化（f（0）= 0，f '（0）1）亚纯单叶函数的第二系数没有一致的上界。因此，球面线性不变性和欧几里得或双曲线性不变性之间存在差异也就不足为奇了。
This paper is a continuation of two earlier papers ([3],[4]) in the sense that in all of these papers we investigate ideas related to linear invariance and uniform local convexity, but in different geometries in the three papers. We assume the reader has some familiarity with these two papers since we sometimes omit details of proofs that are very similar to corresponding proofs in these two preceding papers; in such instances clear reference to the earlier papers is always provided. This paper deals with two notions of spherical linear invariance, one for locally schlicht meromorphic functions from the unit disk D to the Riemann sphere P and another for locally univalent meromorphic functions from the complex plane C to the Riemann sphere. Our two earlier papers deal with linear invariance and uniform local convexity for locally univalent functions from the disk D to either the complex plane C or the unit disk. Here we extend the concept of linear invariance to locally univalent meromorphic functions f: DP orf: C→ P by paralleling the definition of euclidean linear invariance ([11],[3]) rather than that of hyperbolic linear invariance [4]. In the case of euclidean and hyperbolic linear invariance the order of a schlicht function is less than or equal to 2; there is no analogous result for spherical linear invariance because there is no uniform upper bound for the second coefficient of normalized (f (0)= 0, f'(0) 1) meromorphic univalent functions in the unit disk D. Therefore, it should come as no surprise that there are differences between spherical linear invariance and euclidean or hyperbolic linear invariance.