A proof of Hall's conjecture on length of ray images under starlike mappings of order $\alpha$

Assume that $f$ lies in the class of starlike functions of order $\alpha \in [0,1)$,thatis,whichareregularandunivalentfor$|z|<1$andsuchthat$$\mathrm{R}\mathrm{e}\left(\frac{z{f}^{\mathrm{\prime }}(z)}{f(z)}\right)>\alpha for\mathrm{\mid }z\mathrm{\mid }<1.$$Inthispaperweshowthatforeach$\alpha \in [0,1)$, the following sharp inequality holds: |f(re^{i\theta})|^{-1} \int_{0}^{r}|f'(ue^{i\theta})| du \leq \frac{\Gamma (\frac12)\Gamma (2-\alpha )}{\Gamma (\frac32-\alpha )} ~\mbox {for every $r<1$ and $\theta$}. This settles the conjecture of Hall (1980).