Let \(\varphi _{1},\varphi _{2},\varphi _{3}\) be three functions meromorphic in the unit disc \(\Delta \) and continuous on the closure of \(\Delta \) such that \(\varphi _{i} (z)\ne \varphi _{j}(z)\) on the unit circle \(|z|=1.\) Let \(\mathcal {F}\) be a family of meromorphic functions such that \(f \ne \varphi _{j}\) on \(\Delta \) for \(j=1,2,3 ~\text{ and }~f\in \mathcal {F}.\) Then there exists a constant M such that \((1-|z|^{2})f^{\#}(z)\le M \) for each \(z\in \Delta ~\text{ and }~f\in \mathcal {F}.\) In addition, this constant M depends only on the three functions \(\varphi _{1},\varphi _{2}~\text{ and }~\varphi _{3}.\) This generalizes the related theorems due to Lappan (In: Progress in analysis: proceedings of the 3rd international ISAAC congress in progress in analysis, vol 1, pp 221–228. World Scientific, 2003), and Xu and Qiu (C R Math 349:1159–1160, 2011), respectively.

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关于正常功能回避标准的注意事项

假设\（\ varphi _ {1}，\ varphi _ {2}，\ varphi _ {3} \）是单位圆盘\（\ Delta \）中亚纯的三个函数，并且在\（\ Delta \ ）使得\（\ varphi _ {i}（z）\ ne \ varphi _ {j}（z）\）在单位圆\（| z | = 1。\）上使\（\ mathcal {F} \ ）是亚纯函数族，例如\（\ Delta \）上的\（f \ ne \ varphi _ {j} \）对于\（j = 1,2,3〜\ text {和}〜f \ in mathcal {F}。\）然后存在一个常数M，使得每个\（z \ in \ ）的\（（1- | z | ^ {2}）f ^ {\＃}（z）\ le M \）\ mathcal {F}中的Delta〜\ text {和}〜f \。\）此外，此常数M仅取决于三个函数\（\ varphi _ {1}，\ varphi _ {2}〜\ text {和}〜\ varphi _ {3}。\）这归纳了相关定理，因为Lappan（在：分析中的进展：第三届ISAAC国际分析中的大会的议事录，第1卷，第221–228页，世界科学，2003年）和Xu and Qiu（CR Math 349：1159-1160，2011），分别。