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 of order $\alpha \in [0,1)$, that is, which are regular and univalent for $|z|<1$ and such that $${\rm Re} \left (\frac{zf'(z)}{f(z)} \right ) > \alpha ~\mbox{ for } |z|<1. $$ In this paper we show that for each $\alpha \in [0,1)$, the following sharp inequality holds: $$ |f(re^{i\theta})|^{-1} \int_{0}^{r}|f'(ue^{i\theta})| du \leq \frac{\Gamma (\frac12)\Gamma (2-\alpha )}{\Gamma (\frac32-\alpha )} ~\mbox {for every $r<1$ and $\theta$}. $$ This settles the conjecture of Hall (1980).