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Mathematics > Complex Variables

arXiv:1812.08311 (math)
[Submitted on 20 Dec 2018 (v1), last revised 29 Apr 2021 (this version, v2)]

Title:Univalent polynomials and Koebe's one-quarter theorem

Authors:Dmitriy Dmitrishin, Konstantin Dyakonov, Alex Stokolos
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Abstract:The famous Koebe $\frac14$ theorem deals with univalent (i.e., injective) analytic functions $f$ on the unit disk $\mathbb D$. It states that if $f$ is normalized so that $f(0)=0$ and $f'(0)=1$, then the image $f(\mathbb D)$ contains the disk of radius $\frac14$ about the origin, the value $\frac14$ being best possible. Now suppose $f$ is only allowed to range over the univalent polynomials of some fixed degree. What is the optimal radius in the Koebe-type theorem that arises? And for which polynomials is it attained? A plausible conjecture is stated, and the case of small degrees is settled.
Comments: 13 pages
Subjects: Complex Variables (math.CV); Classical Analysis and ODEs (math.CA)
MSC classes: 30C10, 30C25, 30C55, 30C75
Cite as: arXiv:1812.08311 [math.CV]
  (or arXiv:1812.08311v2 [math.CV] for this version)
  https://doi.org/10.48550/arXiv.1812.08311
Journal reference: Anal. Math. Phys. 9 (2019), no. 3, 991--1004
Related DOI: https://doi.org/10.1007/s13324-019-00305-x

Submission history

From: Konstantin Dyakonov [view email]
[v1] Thu, 20 Dec 2018 01:54:54 UTC (13 KB)
[v2] Thu, 29 Apr 2021 04:47:57 UTC (13 KB)
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