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An alternate proof of Hall's theorem on a conformal mapping inequality

1996
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Bulletin of the Belgian Mathematical Society Simon Stevin
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In this note we give a different and direct proof of the following result of Hall [2] , which actually implies the conjecture of Sheil-Small [3] . For details about the related problems we refer to [1, 3] . THEOREM. Let f be regular for |z| < 1 and f(0) = 0. Further, let f be starlike of order 1/2. Then r 0 |f (ρe iθ )|dρ < π 2 |f(re iθ )| for every r < 1 and real θ. Proof. As in [2, p.125] (see also [1]), to prove our result it suffices to show that J = I(t, τ ) + I(τ, t) < π − 2 for 0 < t < τ

doi:10.36045/bbms/1105540793
fatcat:qco4pfeul5aglclgjt5anwxglm