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


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