A proof of Hall's conjecture on length of ray images under starlike mappings of order α

Abstract

Assume that \(f\) lies in the class of starlike functions of order \(\alpha\in[0,1)\), that is, which are regular and univalent for \(|z|<1\) and such that Re\(\left(\frac{zf'(z)}{f(z)}\right)>\alpha\) for \(|z|<1.\) In this paper we show that for each \(\alpha\in[0,1)\), the following sharp inequality holds: \(|f(re^{i\theta})|^{-1}\int_{0}^{r}|f'(ue^{i\theta})|\,du\leq\frac{\Gamma(\frac{1}{2})\Gamma(2-\alpha)}{\Gamma(\frac{3}{2}-\alpha)}\) for every \(r<1\) and \(\theta\). This settles the conjecture of Hall (1980) positively.