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

Authors

  • Peter Hästö University of Turku, Department of Mathematics and Statistics
  • Saminathan Ponnusamy Indian Institute of Technology Madras, Department of Mathematics

Keywords:

Ray-image, length of ray-image, starlike and univalent mappings, starlike functions of order α

Abstract

 

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.
Section
Articles

Published

2022-01-24

How to Cite

Hästö, P., & Ponnusamy, S. (2022). A proof of Hall’s conjecture on length of ray images under starlike mappings of order α. Annales Fennici Mathematici, 47(1), 335–349. https://doi.org/10.54330/afm.113736