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The length of ray-images under starlike mappings
Published online by Cambridge University Press: 26 February 2010
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Let w = f(z) be regular and schlicht for |z|, and f (0) = 0.
Suppose that f maps the unit disc {z : |z| < 1} onto a domain D starlike with respect to w = 0. Let C(r, θ) be the image in D of the ray joining z = 0 to z = reiθ, and let
be its length. Sheil–Small [1] proved that l(r, θ) < (1 + log 4) | f (reiθ)|, and conjectured the following result, which it is my aim to prove in this paper.
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