On the Koebe Quarter Theorem for certain polynomials

Ignaciuk, Szymon; Parol, Maciej

doi:10.1007/s13324-021-00501-8

Cited by 6 publications

(4 citation statements)

References 8 publications

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“…and n = 3. Note that in contrast with the cases n = 1, 2, the problem of determining the extremal properties of polynomials from different classes even for small n ≥ 3 is usually far from trivial [17].…”

Section: Problem Statementmentioning

confidence: 99%

Extremal problems for trinomials with fold symmetry

Dmitrishin¹,

Stokolos²,

Gray³

2022

Preprint

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The famous T. Suffridge polynomials have many extremal properties: the maximality of coefficients when the leading coefficient is maximal; the zeros of the derivative are located on the unit circle; the maximum radius of stretching the unit disk with the schlicht normalization F (0) = 0, F ′ (0) = 1; the maximum size of the unit disk contraction in the direction of the real axis for univalent polynomials with the normalization F (0) = 0, F (1) = 1. However, under the standard symmetrization method T F (z T ), these polynomials go to functions, which are not polynomials. How can we construct the polynomials with fold symmetry that have properties similar to those of the Suffridge polynomial? What values will the corresponding extremal quantities take in the above-mentioned extremal problems? The paper is devoted to solving these questions for the case of the trinomials F (z) = z + az 1+T + bz 1+2T . Also, there are suggested hypotheses for the general case in the work.

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Section: Problem Statementmentioning

confidence: 99%

Extremal problems for trinomials with fold symmetry

Dmitrishin¹,

Stokolos²,

Gray³

2022

Preprint

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“…Classical problems of geometric complex analysis related to extremal stretching and contraction of the unit disk D by typically real polynomials were solved in [2,[4][5][6]15] (the extreme values and corresponding extremizers were found). Let us also note that various extremal problems in subclasses of typically real or univalent polynomials were considered, for example, in [12,[20][21][22].…”

Section: Introductionmentioning

confidence: 99%

“…In the following, we need the maximal root of the equation ∆ N = 0 (see (12) for ∆ N ). Denote it by η N , and the corresponding eigenvector by Z (1) (η N ).…”

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confidence: 99%

An extremal problem for polynomials

Dmitrishin

Smorodin

Stokolos

2022

Applied and Computational Harmonic Analysis

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“…This radius is called the Koebe radius. In [9][10][11]13], some examples of finding the Koebe radius for different classes of functions are given.…”

Section: Introductionmentioning

confidence: 99%

Some extremal problems for trinomials with fold symmetry

2022

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The Koebe problem for univalent polynomials with real coefficients is fully solved only for trinomials, which means that in this case the Koebe radius and the extremal polynomial (extremizer) have been found. The general case remains open, but conjectures have been formulated. The corresponding conjectures have also been hypothesized for univalent polynomials with real coefficients and T -fold rotational symmetry. This paper provides confirmation of these hypotheses for trinomials z +az T +1 +bz 2T +1 . Namely, the Koebe radius is r = 4 cos 2 π(1+T ) 2+3T , and the only extremizer of the Koebe problem is the trinomial

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On the Koebe Quarter Theorem for certain polynomials

Abstract: We study problems similar to the Koebe Quarter Theorem for close-to-convex polynomials with all zeros of derivative in $${\mathbb {T}}:=\{z\in {\mathbb {C}}:|z|=1\}$$ T : = { z ∈ C : | z | = 1 } … Show more

Cited by 6 publications

References 8 publications

Extremal problems for trinomials with fold symmetry

Extremal problems for trinomials with fold symmetry

An extremal problem for polynomials

Some extremal problems for trinomials with fold symmetry

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