Uber diese Arbeit wurde auf eiriern “Tag der Funktionentheorie“ am 15. und 16.6.1984 an der RWTH Aachen berichtet. Let Ω denote the set of continuous functions Q:[0,1)→R such that (a) (1-x2)2Q(x) is nonincreasing and (b) the solution of the initial value problem y'' +Q(x)y=0, y(0)=1, y'(0)=0 is Positive in [0, 1). We prove the following theorem: Let f be locally univalent in the unit disk D:|z|<1, and assume |Sf|≤2Q(|z|), z∈D, for some Q∈Ω, where Sf denotes the Schwarzian derivative of f Then f has a homeomorphic extension to D˜, except when Q is real-analytic and ∫10(dx)/[y2(x)] diverges. In this case ∂f(D) may have a double point.
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