Asymptotic perimeter estimates for spirallike functions

Abstract

By a result of F. R. Keogh in 1959, we have the inequality L(r,f)/M(r,f)≤ 2π+4log[(1+r)/(1−r)] for a starlike function f on the unit disk, where L(r,f) and M(r,f) denote the length of the curve θ↦ f(re^iθ) and the maximum value of |f(z)| on the circle |z|=r for 0<r<1. E. Crane and D. Markose in their 2005 paper showed that the constant $4$ is best possible. D. K. Thomas obtained in 1968 the inequality L(r,f)/√(A(r,f))≤ 2√π(1+log[(1+r)/(1−r)]) for a starlike function f, where A(r,f) is the area of the image of the disk |z|<r under the mapping f. In this paper, we extend these results to the case of λ-spirallike functions for a given λ with −π/2<λ<π/2.

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BibTeX RIS

BibTeX

@misc{alsharari2026asymptotic,
  title = {{Asymptotic perimeter estimates for spirallike functions}},
  author = {Zain Shayim Alsharari and Toshiyuki Sugawa},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24035},
  eprint = {2609.24035},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24035},
}

RIS

TY  - UNPB
TI  - Asymptotic perimeter estimates for spirallike functions
AU  - Zain Shayim Alsharari
AU  - Toshiyuki Sugawa
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24035
UR  - https://arxiv.org/abs/2609.24035
AB  - By a result of F. R. Keogh in 1959, we have the inequality L(r,f)/M(r,f)≤ 2π+4log[(1+r)/(1−r)] for a starlike function f on the unit disk, where L(r,f) and M(r,f) denote the length of the curve θ↦ f(re^iθ) and the maximum value of |f(z)| on the circle |z|=r for 0<r<1. E. Crane and D. Markose in their 2005 paper showed that the constant $4$ is best possible. D. K. Thomas obtained in 1968 the inequality L(r,f)/√(A(r,f))≤ 2√π(1+log[(1+r)/(1−r)]) for a starlike function f, where A(r,f) is the area of the image of the disk |z|<r under the mapping f. In this paper, we extend these results to the case of λ-spirallike functions for a given λ with −π/2<λ<π/2.
ER  -

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