Let \(z_0\) and \(w_0\) be given points in the open unit disk \(\mathbb {D}\) with \(|w_0| < |z_0|\), and \(\mathcal {H}_0\) be the class of all analytic self-maps f of \(\mathbb {D}\) normalized by \(f(0)=0\). In this paper, we establish the third-order Dieudonné’s Lemma and apply it to explicitly determine the variability region \(\{f'''(z_0): f\in \mathcal {H}_0,f(z_0) =w_0, f'(z_0)=w_1\}\) for given \(z_0,w_0,w_1\) and give the form of all the extremal functions.

中文翻译：

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有界解析函数的三阶导数的可变域

让\(z_0\)和\(w_0\)被赋予开放单位盘\(\mathbb {D}\) 中的点，其中\(|w_0| < |z_0|\)和\(\mathcal {H} _0 \）是该类的所有解析自映射的˚F的\（\ mathbb {d} \）通过标准化\（F（0）= 0 \） 。在本文中，我们建立了三阶 Dieudonné 引理并将其应用于明确的确定变异区域\(\{f'''(z_0): f\in \mathcal {H}_0,f(z_0) =w_0, f'(z_0)=w_1\}\)对于给定的\(z_0,w_0,w_1\)并给出所有极值函数的形式。