The class of holomorphic self-maps of a disk with a boundary fixed point is studied. For this class of functions, the famous Julia–Carathéodory theorem gives a sharp estimate of the angular derivative at the boundary fixed point in terms of the image of the interior point. In the case when additional information about the value of the derivative at the interior point is known, a sharp estimate of the angular derivative at the boundary fixed point is obtained. As a consequence, the sharpness of the boundary Dieudonné–Pick lemma is established and the class of the extremal functions is identified. An unimprovable strengthening of the Osserman general boundary lemma is also obtained.

中文翻译：

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在边界上Dieudonné–Pick Lemma

研究了具有边界固定点的磁盘的全纯自映射的类。对于此类函数，著名的Julia-Carathéodory定理就内点图像而言，对边界固定点处的角导数给出了清晰的估计。在已知关于内点处的导数值的附加信息的情况下，可以获得边界固定点处的角导数的清晰估计。结果，确定了边界Dieudonné-Pick引理的清晰度，并确定了极值函数的类别。还获得了Osserman一般边界引理的不可改进的加强。