论文标题
霍尔在订单$α$的星光映射下的射线图像长度的猜想证明
A proof of Hall's conjecture on length of ray images under starlike mappings of order $α$
论文作者
论文摘要
假设$ f $在于[0,1）$中的顺序$α\的恒星函数，即$ | z | <1 $且无需$ | z | <1 $，因此，$ | $$在本文中，我们表明，对于[0,1）$中的每个$α\，以下尖锐的不平等存在：$$ | f（re^{iθ}）|^{ - 1} \ int_ {0}^{r}^{r} {r} {r} | f'（ue^{iθ}}}}} | du \ leq \ frac {γ（\ frac12）γ（2-α）}} {γ（\ frac32-α）}〜\ mbox {对于每个$ r <1 $和$θ$}。 $$这解决了霍尔的猜想（1980）。
Assume that $f$ lies in the class of starlike functions of order $α\in [0,1)$, that is, which are regular and univalent for $|z|<1$ and such that $${\rm Re} \left (\frac{zf'(z)}{f(z)} \right ) > α~\mbox{ for } |z|<1. $$ In this paper we show that for each $α\in [0,1)$, the following sharp inequality holds: $$ |f(re^{iθ})|^{-1} \int_{0}^{r}|f'(ue^{iθ})| du \leq \frac{Γ(\frac12)Γ(2-α)}{Γ(\frac32-α)} ~\mbox {for every $r<1$ and $θ$}. $$ This settles the conjecture of Hall (1980).
