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In complex analysis, a branch of mathematics, the Koebe 1/4 theorem states the following: Koebe Quarter Theorem. The image of an injective analytic function ''f'' : D → C from the unit disk D onto a subset of the complex plane contains the disk whose center is ''f''(0) and whose radius is |''f′''(0)|/4. The theorem is named after Paul Koebe, who conjectured the result in 1907. The theorem was proven by Ludwig Bieberbach in 1916. The example of the Koebe function shows that the constant 1/4 in the theorem cannot be improved. A related result is the Schwarz lemma, and a notion related to both is conformal radius. ==Grönwall's area theorem== Suppose that : is univalent in |''z''| > 1. Then : In fact, if ''r'' > 1, the complement of the image of the disk ''|z|'' > ''r'' is a bounded domain ''X''(''r''). Its area is given by : Since the area is positive, the result follows by letting ''r'' decrease to 1. The above proof shows equality holds if and only if the complement of the image of ''g'' has zero area, i.e. Lebesgue measure zero. This result was proved in 1914 by the Swedish mathematician Thomas Hakon Grönwall. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「In complex analysis, a branch of mathematics, the Koebe 1/4 theorem states the following:Koebe Quarter Theorem. The image of an injective analytic function ''f'' : D → C from the unit disk D onto a subset of the complex plane contains the disk whose center is ''f''(0) and whose radius is |''f′''(0)|/4.The theorem is named after Paul Koebe, who conjectured the result in 1907. The theorem was proven by Ludwig Bieberbach in 1916. The example of the Koebe function shows that the constant 1/4 in the theorem cannot be improved.A related result is the Schwarz lemma, and a notion related to both is conformal radius.==Grönwall's area theorem==Suppose that :g(z) = z +b_1z^ + b_2 z^ + \cdotsis univalent in |''z''| > 1. Then:\sum_ n|b_n|^2 \le 1.In fact, if ''r'' > 1, the complement of the image of the disk ''|z|'' > ''r'' is a bounded domain ''X''(''r''). Its area is given by: \int_ dx\,dy = \int_\overline\,dz = -\int_\overline\,dg= -\sum n|b_n|^2 r^.Since the area is positive, the result follows by letting ''r'' decrease to 1. The above proof shows equality holds if and only if the complement of the image of ''g'' has zero area, i.e. Lebesgue measure zero.This result was proved in 1914 by the Swedish mathematician Thomas Hakon Grönwall.」の詳細全文を読む スポンサード リンク
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