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In complex analysis, de Branges's theorem, or the Bieberbach conjecture, is a theorem that gives a necessary condition on a holomorphic function in order for it to map the open unit disk of the complex plane injectively to the complex plane. It was posed by and finally proven by . The statement concerns the Taylor coefficients ''an'' of such a function, normalized as is always possible so that ''a''0 = 0 and ''a''1 = 1. That is, we consider a function defined on the open unit disk which is holomorphic and injective (''univalent'') with Taylor series of the form : such functions are called ''schlicht''. The theorem then states that : ==Schlicht functions== The normalizations :''a''0 = 0 and ''a''1 = 1 mean that :''f''(0) = 0 and ''f'' '(0) = 1; this can always be assured by a linear fractional transformation: starting with an arbitrary injective holomorphic function ''g'' defined on the open unit disk and setting : Such functions ''g'' are of interest because they appear in the Riemann mapping theorem. A schlicht function is defined as an analytic function ''f'' that is one-to-one and satisfies ''f''(0) = 0 and ''f'' '(0) = 1. A family of schlicht functions are the rotated Koebe functions : with α a complex number of absolute value 1. If ''f'' is a schlicht function and |''a''''n''| = ''n'' for some ''n'' ≥ 2, then ''f'' is a rotated Koebe function. The condition of de Branges' theorem is not sufficient to show the function is schlicht, as the function : shows: it is holomorphic on the unit disc and satisfies |''a''''n''|≤''n'' for all ''n'', but it is not injective since ''f''(−1/2 + ''z'') = ''f''(−1/2 − ''z''). 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「De Branges's theorem」の詳細全文を読む スポンサード リンク
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