Sharp Subordination Radii through Boundary Minimization
Abstract
Let φ and ψ be univalent functions in the unit disk 𝔻 satisfying φ(0)=ψ(0)=1 and φnot≺ψ, and let 𝒫φ denote the class of analytic functions p in 𝔻 such that p≺φ. Under suitable analytic continuation, univalence, and boundary-behavior hypotheses on the branch of φ−1 satisfying φ−1(1)=0, we determine the sharp 𝒫ψ-radius of 𝒫φ as ℛ(φ,ψ) = min|ζ|=1 | φ−1(ψ(ζ)) |. For Ma--Minda functions satisfying these hypotheses, the same radius is sharp for the associated classes 𝒮𝒯(φ) and 𝒞𝒱(φ) with respect to 𝒮𝒯(ψ) and 𝒞𝒱(ψ), respectively. We apply the general result to $ φ_{\mathrm L}(z)=\sqrt{1+z}, φ_{\mathrm{\leftmoon}}(z)=z+\sqrt{1+z^2}, φ_{\mathrm{Lim}}(z)=\left(1+{z}/{\sqrt2}\right)^2, $ for several choices of the Ma--Minda function ψ, and obtain parameter-dependent extensions for the corresponding generalized families. The resulting boundary minima are evaluated analytically.
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BibTeX
@misc{palanisamy2026sharp,
title = {{Sharp Subordination Radii through Boundary Minimization}},
author = {Poomalai Palanisamy and Ravichandran Vaithiyanathan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12006},
eprint = {2610.12006},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12006},
}
RIS
TY - UNPB
TI - Sharp Subordination Radii through Boundary Minimization
AU - Poomalai Palanisamy
AU - Ravichandran Vaithiyanathan
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.12006
UR - https://arxiv.org/abs/2610.12006
AB - Let φ and ψ be univalent functions in the unit disk 𝔻 satisfying φ(0)=ψ(0)=1 and φnot≺ψ, and let 𝒫_φ denote the class of analytic functions p in 𝔻 such that p≺φ. Under suitable analytic continuation, univalence, and boundary-behavior hypotheses on the branch of φ^−1 satisfying φ^−1(1)=0, we determine the sharp 𝒫_ψ-radius of 𝒫_φ as ℛ(φ,ψ) = min_|ζ|=1 | φ^−1(ψ(ζ)) |. For Ma--Minda functions satisfying these hypotheses, the same radius is sharp for the associated classes 𝒮𝒯(φ) and 𝒞𝒱(φ) with respect to 𝒮𝒯(ψ) and 𝒞𝒱(ψ), respectively. We apply the general result to $ φ_{\mathrm L}(z)=\sqrt{1+z}, φ_{\mathrm{\leftmoon}}(z)=z+\sqrt{1+z^2}, φ_{\mathrm{Lim}}(z)=\left(1+{z}/{\sqrt2}\right)^2, $ for several choices of the Ma--Minda function ψ, and obtain parameter-dependent extensions for the corresponding generalized families. The resulting boundary minima are evaluated analytically.
ER -