Geodesic
A proof of Hall's conjecture on length of ray images under starlike mappings of order α
Peter Hästö1 ; Saminathan Ponnusamy2
1University of Turku, Department of Mathematics and Statistics
2Indian Institute of Technology Madras, Department of Mathematics
Annales Fennici Mathematici, Tome 47 (2022) no. 1, pp. 335-349

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  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.
DOI : 10.54330/afm.113736
Keywords: Ray-image, length of ray-image, starlike and univalent mappings, starlike functions of order α

Peter Hästö  1   ; Saminathan Ponnusamy  2

1 University of Turku, Department of Mathematics and Statistics
2 Indian Institute of Technology Madras, Department of Mathematics 
Peter Hästö; Saminathan Ponnusamy. A proof of Hall's conjecture on length of ray images under starlike mappings of order α. Annales Fennici Mathematici, Tome 47 (2022) no. 1, pp. 335-349. doi: 10.54330/afm.113736
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