In the Jack’s lemma it is considered q(z), an analytic function in \(|z|<1\) with \(q(0)=0\) for which |q(z)| attains its maximum value on the disc \(|z|\le r<1\) at the point \(z_0\), \(|z_0|=r\). Then \(z_0q'(z_0)=kq(z_0)\) and \(k\ge 1\). In this paper we try to say more about the number k in a generalizations of this lemma, where we consider \(\max |\arg \{q(z)\}|\) or \(\min |\mathfrak {Re} \{q(z)\}|\) instead of |q(z)|. In these cases q(z) has another normalization at the origin.

中文翻译：

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杰克引理的应用

在杰克引理中，它被认为是\(|z|<1\)中的解析函数q ( z ) ，其中\(q(0)=0\) ，其中 | q ( z )|在盘\(|z|\le r<1\)点\(z_0\)、\(|z_0|=r\)处达到其最大值。然后\(z_0q'(z_0)=kq(z_0)\)和\(k\ge 1\)。在本文中，我们尝试在这个引理的概括中更多地讨论数字k ，其中我们考虑\(\max |\arg \{q(z)\}|\)或\(\min |\mathfrak {Re } \{q(z)\}|\)而不是 | q ( z )|。在这些情况下，q ( z ) 在原点有另一个归一化。