The Carathéodory metric in plane domains
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概要
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Let <I>D</I>∉<I>O</I><SUB><I>AB</I></SUB> be a plane domain and let <I>C</I><SUB><I>D</I></SUB>(<I>z</I>) be its analytic capacity at <I>z</I>∈<I>D</I>. Let \mathscr{K}<SUB><I>D</I></SUB>(<I>z</I>) be the curvature of the Carathéodory metric <I>C</I><SUB><I>D</I></SUB>(<I>z</I>)<I>|dz|</I>. We show that \mathscr{K}<SUB><I>D</I></SUB>(<I>z</I>)<−4 the Ahlfors function of <I>D</I> with respect to <I>z</I> has a zero other than <I>z</I>. For finite <I>D</I>, \mathscr{K}<SUB><I>D</I></SUB>(<I>z</I>){≤}−4 and equality holds if and only if <I>D</I> is simply connected. As a corollary we obtain a result proved first by Suita, namely, that \mathscr{K}<SUB><I>D</I></SUB>(<I>z</I>){≤}−4 if <I>D</I>∉<I>O</I><SUB><I>AB</I></SUB>. Several other properties related to the Carathéodory metric are proven.
- 国立大学法人 東京工業大学大学院理工学研究科数学専攻の論文
国立大学法人 東京工業大学大学院理工学研究科数学専攻 | 論文
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