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

R. Balasubramanian, S. Ponnusamy
1996 Bulletin of the Belgian Mathematical Society Simon Stevin  
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
more » ... π − 2 for 0 < t < τ < π where To evaluate these integrals we define k by k 2 = sin 2 (τ /2) − sin 2 (t/2) cos 2 (t/2) sin 2 (τ /2)
doi:10.36045/bbms/1105540793 fatcat:qco4pfeul5aglclgjt5anwxglm