Let $D \subset \mathbb{C}$ be a domain with $0 \in D$. For $R>0$, let ${{\hat \omega }_D}\left( {R} \right)$ denote the harmonic measure of $ D \cap \left\{ {\left| z \right| = R} \right\}$ at $0$ with respect to the domain $ D \cap \left\{ {\left| z \right| 0$. Thus, the arising question, first posed by Betsakos, is the following: Does there exist a positive constant $C$ such that for all simply connected domains $D$ with $0 \in D$ and all $R>0$, \[{\omega_D}\left( {R} \right) \ge C{{\hat \omega }_D}\left( {R} \right)? \] In general, we prove that the answer is negative by means of two different counter-examples. However, under additional assumptions involving the geometry of $D$, we prove that the answer is positive. We also find the value of the optimal constant for starlike domains.

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关于简单连通域的谐波测度的性质

令 $D \subset \mathbb{C}$ 是一个 $0 \in D$ 的域。对于$R>0$，令${{\hat \omega }_D}\left( {R} \right)$ 表示$ D \cap \left\{ {\left| z \右| = R} \right\}$ 在 $0$ 相对于域 $ D \cap \left\{ {\left| z \右| 0 美元。因此，Betsakos 首先提出的问题如下：是否存在正常数 $C$ 使得对于所有具有 $0 \in D$ 且所有 $R>0$ 的单连通域 $D$，\[ {\omega_D}\left( {R} \right) \ge C{{\hat \omega }_D}\left( {R} \right)? \] 一般来说，我们通过两个不同的反例来证明答案是否定的。然而，在涉及 $D$ 几何的额外假设下，我们证明答案是肯定的。我们还找到了星状域的最佳常数值。