Applications of hyperbolic convexity to euclidean and spherical convexity
Applications of hyperbolic convexity to euclidean and spherical convexity
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双曲凸性在欧氏凸性和球凸性中的应用
DOI：
10.1007/bf02792893
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发表时间：
1987
期刊：
影响因子：
--
通讯作者：
D. Minda
中科院分区：
文献类型：
--
作者：
D. Minda
文[8]建立了双曲度规的反射原理。这个反射原理的一个应用是双曲凸性的一个充分条件。特别地，如果双曲平面区域~关于f2的闭包中的某点a是星形的，并且A是中心为a的任何圆盘，则f2 AA是作为fL的子集的双曲凸的。特别地，如果s是复平面C中的欧几里德凸区域，则f2 f~A对于中心为a的任何圆盘A是双曲凸的。后一个结果是尖锐的：如果f2是一个半平面，A是一个中心不在el（f2）中的圆盘，那么~f~A不是双曲凸的。这些双曲凸的结果导致应用程序的区域是凸的，无论是欧几里德或球面几何。例如，如果f~是C中的凸区域，y是f~2中的双曲测地线，则y的任何曲率圆的中心都不可能位于~中。这种类型的其他结果是已知的。J~rgensen [5]指出，对于C中的一般双曲区域，曲率圆必须与边界相交。Osgood [10]观察到，内塔尼亚胡[9]的一个结果将这一点改进为单连通区域，即z0与z 0处y的曲率圆心之间的距离必须至少为] 8a（z0），其中~（z0）表示从z0到af 2的欧几里得距离。我们的结果的曲率圆的中心导致几个尖锐的失真定理的双曲度量，完善已知的事实，单连通区域。例如，对于凸区域，IV~（z）l <l/8n（z），其中2~表示f2上双曲度量的密度，等式成立当且仅当f~是半平面。这在单叶函数理论中有一个结果。若fEK是单位圆盘上的正规化凸单叶函数类，则f（z）= z/（1-e~）（0）If "（0）[<1且等式成立当且仅当f（z）= z/（1-e~）对某个0 ∈ R.对于规范化单叶函数的完整类S，内塔尼亚胡[9]证明了类似的尖锐上界是~，并且Koebe函数不是极值。最后，我们考虑黎曼球面P上的区域是凸的球面几何，并获得类似的所有我们的结果关于欧几里德凸区域。这些导致所有单叶函数类Ks（a）的应用90
In [8] a reflection principle for the hyperbolic metric was established. One application of this reflection principle was a sufficient condition for hyperbolic convexity. In particular, if a hyperbolic plane region~ is starlike with respect to some point a in the closure of f2 and A is any disk with center a, then f2 AA is hyperbolically convex as a subset of fL In particular, if s is a euclidean convex region in the complex plane C, then f2 f~ A is hyperbolically convex for any disk A with center a in el (f2). This latter result is sharp: if f2 is a half-plane and A is a disk which does not have its center in el (f2), then~ f~ A is not hyperbolically convex. These results on hyperbolic convexity lead to applications for regions that are convex in either euclidean or spherical geometry. For example, if f~ is a convex region in C and y is a hyperbolic geodesic in f2, then the center of any circle of curvature for y cannot lie in~. Other results of this type are known. J~ rgensen [5] noted that for a general hyperbolic region in C the circle of curvature must intersect the boundary. Osgood [10] observed that a result of Netanyahu [9] refined this for simply connected regions to the distance between z0 and the center of the circle of curvature for y at z0 must be at least] 8a (zo), where~(z0) denotes the euclidean distance from z0 to af2. Our result on the center of the circle of curvature leads to several sharp distortion theorems for the hyperbolic metric which refine known facts for simply connected regions. For instance, IV~(z) l< l/8n (z) for a convex region~, where 2~ denotes the density of the hyperbolic metric on f2, and equality holds if and only if f~ is a half-plane. This has a consequence in univalent function theory. If fEK, the class of normalized convex univalent functions in the unit disk, then~/~ D)(0) If"(0)[< 1 with equality if and only if f (z)= z/(1-e~~ for some 0 E R. For the full class S of normalized univalent functions Netanyahu [9] demonstrated that the analogous sharp upper bound is~ and that the Koebe function is not extremal. Finally, we consider regions on the Riemann sphere P which are convex in spherical geometry and obtain analogs of all of our results about euclidean convex regions. These lead to an application for the class Ks (a) of all univalent functions 90