Univalent polynomials and Koebe’s one-quarter theorem

Dmitriy Dmitrishin; Konstantin Dyakonov; Alex Stokolos

doi:10.1007/s13324-019-00305-x

Univalent polynomials and Koebe’s one-quarter theorem

Dmitriy Dmitrishin, Konstantin Dyakonov, Alex Stokolos

Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

Abstract

The famous Koebe 14 theorem deals with univalent (i.e., injective) analytic functions f on the unit disk D. It states that if f is normalized so that f(0) = 0 and f^′(0) = 1 , then the image f(D) contains the disk of radius 14 about the origin, the value 14 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.

Original language	English
Pages (from-to)	991-1004
Number of pages	14
Journal	Analysis and Mathematical Physics
Volume	9
Issue number	3
DOIs	https://doi.org/10.1007/s13324-019-00305-x
State	Published - Sep 1 2019

Scopus Subject Areas

Analysis
Algebra and Number Theory
Mathematical Physics

Keywords

Koebe radius
Koebe’s one-quarter theorem
Univalent polynomial

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10.1007/s13324-019-00305-x

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abstract = "The famous Koebe 14 theorem deals with univalent (i.e., injective) analytic functions f on the unit disk D. It states that if f is normalized so that f(0) = 0 and f′(0) = 1 , then the image f(D) contains the disk of radius 14 about the origin, the value 14 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.",

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AB - The famous Koebe 14 theorem deals with univalent (i.e., injective) analytic functions f on the unit disk D. It states that if f is normalized so that f(0) = 0 and f′(0) = 1 , then the image f(D) contains the disk of radius 14 about the origin, the value 14 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.

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Univalent polynomials and Koebe’s one-quarter theorem

Abstract

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Keywords

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