Some results on univalent functions with positive Hayman index (Q2769215)

Let \(f\) denote a univalent function in the class \(S\) with positive Hayman index \(\alpha\) and as direction of maximal growth the positive \(x\)-axis. Let \(k\) denote the Koebe function. Hu and Dong, in 1987, proved that \(f/k\) belongs to all the Hardy spaces \(H^p\) for \(p< \infty,\) but need not belong to BMOA. In the paper under review, Girela proves that if in addition \(f\) is close-to-convex, then \(f/k\) does belong to BMOA, but need not be bounded. The proof of the positive result uses the method of star-functions. Girela further complements the Hu-Dong results by showing that if \(f\) has positive Hayman index and is not necessarily close-to-convex, then \(k/f\) is in all the \(H^p\) for \(p<\infty.\) Left open are the questions of whether \(k/f\) belongs to \(H^{\infty}\) or to BMOA.