On Phragmén-Lindelöf’s principle
On Phragmén-Lindelöf’s principle
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关于 Phragmén-Lindelöf 原理
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发表时间：
1937
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通讯作者：
L. Ahlfors
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作者：
L. Ahlfors
在本文的第一部分中，我们给出了 Phragmen-Lindelkf 现在的经典原理的证明，该证明比迄今为止已知的任何证明都更简单，并且产生了更详细的信息。我们的程序包括用有限项证明一个定理，类似于哈达玛的三圆定理，从中可以看出渐近陈述遵循非常简单和透明的推理。通过允许被认为具有两个可能的奇点的函数获得结果的某种对称性，一个在 0 处，一个在 oo 处。最终定理是最清晰的，包含所有先前的结果，包括 Nevanlinna 兄弟的结果。在第二部分中，我们将 Phragmen-Lindel6f 的原理推广到 n 个变量的调和函数。第一部分的方法被认为可以毫无困难地延续下去。结果特别有趣，因为二维情况的对称性没有得到维持，对应于两个奇点的极值函数现在本质上不同。
In the first part of this paper we give a proof of Phragmen-Lindelkf's now classical principlet which is simpler and yields more detailed information than any of the proofs hitherto known. Our procedure consists in proving a theorem in finite terms, similar to Hadamard's three circles theorem, from which the asymptotic statement is shown to follow by a very simple and transparent reasoning. A certain symmetry in the result is obtained by allowing the functions considered to have two possible singularities, one at 0 and one at oo. The ultimate theorem is the sharpest possible and contains all previous results, including those of the brothers Nevanlinna.t In Part II we generalize Phragmen-Lindel6f's principle to harmonic functions of n variables. The methods of Part I are seen to carry over without any difficulties. The result is particularly interesting in so far as the symmetry of the two-dimensional case is not maintained, the extremal functions corresponding to the two singularities being now essentially different.