Radii of Starlikeness of Spirallike Functions
Abstract
A normalized analytic function f defined on the unit disk 𝔻 is called α-spirallike if f(𝔻) is an α-spirallike domain, that is, for every z∈ f(𝔻), the α-spiral ze−eiαt, t≥0, is contained in f(𝔻), or equivalently, if Re (eiα zf'(z)/f(z))>0 for all z∈ 𝔻. The 0-spirallike functions are the usual starlike functions. The function f is parabolic starlike, if Re (zf'(z)/f(z))>|zf'(z)/f(z)−1|, and lemniscate starlike if |(zf'(z)/f(z))2−1|<1 respectively. In this paper, we determine the radii of parabolic starlikeness and lemniscate starlikeness of α-spirallike functions. Our approach combines geometric characterizations of the underlying regions with the resultant method for eliminating auxiliary variables, leading to explicit equations for the sharp radii. Some known radius results are recovered as special cases.
Related
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BibTeX
@misc{arul2026radii,
title = {{Radii of Starlikeness of Spirallike Functions}},
author = {Lakshmipriya Arul and Vaithiyanathan Ravichandran},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06175},
eprint = {2610.06175},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06175},
}
RIS
TY - UNPB
TI - Radii of Starlikeness of Spirallike Functions
AU - Lakshmipriya Arul
AU - Vaithiyanathan Ravichandran
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.06175
UR - https://arxiv.org/abs/2610.06175
AB - A normalized analytic function f defined on the unit disk 𝔻 is called α-spirallike if f(𝔻) is an α-spirallike domain, that is, for every z∈ f(𝔻), the α-spiral ze^−e^iαt, t≥0, is contained in f(𝔻), or equivalently, if Re (e^iα zf'(z)/f(z))>0 for all z∈ 𝔻. The 0-spirallike functions are the usual starlike functions. The function f is parabolic starlike, if Re (zf'(z)/f(z))>|zf'(z)/f(z)−1|, and lemniscate starlike if |(zf'(z)/f(z))^2−1|<1 respectively. In this paper, we determine the radii of parabolic starlikeness and lemniscate starlikeness of α-spirallike functions. Our approach combines geometric characterizations of the underlying regions with the resultant method for eliminating auxiliary variables, leading to explicit equations for the sharp radii. Some known radius results are recovered as special cases.
ER -