The range of the residue functional for the class \(S_ p\) (Q1081720)

Let \(S_ p\) with \(0<p<1\) denote the class of analytic functions g which are meromorphic and univalent in the unit disc and are normalized by \(g(0)=0\), \(g'(0)=1\) and \(g(p)=\infty\). Observing the residue of \(g\in S_ p\) at \(z=p\), the author shows that the range \(\Omega_ p\) of the residue functional is given by \(\{-p^ 2(1-p^ 2)^{\epsilon}:| \epsilon | \leq 1\}\). The proof depends on the Grunsky inequalities with respect to the familiar classes S and \(\Sigma\).