Abstract Let f be a holomorphic, or even meromorphic, function on the unit disc. Plessner's theorem then says that, for almost every boundary point ζ, either (i) f has a finite nontangential limit at ζ, or (ii) the image f ( S ) of any Stolz angle S at ζ is dense in the complex plane. This paper shows that statement (ii) can be replaced by a much stronger assertion. This new theorem and its analogue for harmonic functions on halfspaces also strengthen classical results of Spencer, Stein and Carleson.

中文翻译：

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普莱斯纳定理的强形式

摘要 设 f 是单位圆盘上的全纯函数，甚至是亚纯函数。然后 Plessner 定理说，对于几乎每个边界点 ζ，要么 (i) f 在 ζ 处有一个有限的非切线极限，要么 (ii) 在 ζ 处的任何 Stolz 角 S 的图像 f ( S ) 在复平面中是稠密的。这篇论文表明，陈述 (ii) 可以用更强的断言代替。这个新定理及其对半空间调和函数的模拟也加强了 Spencer、Stein 和 Carleson 的经典结果。