Extremal Problem for the Area of the Image of a Disk

Salimov, Ruslan; Klishchuk, B. A.

doi:10.1007/s10958-018-4015-6

Cited by 4 publications

(3 citation statements)

References 4 publications

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“…The theory of ring Q-homeomorphisms for p = n was studied in [19,22,23,34], for 1 ă p ă n in [6][7][8][25][26][27] and for p ą n in [11,[28][29][30][31][32][33]. This theory can be applied to the mappings of finite distortion belonging to the Orlicz-Sobolev classes W 1,φ loc under the Calderon condition, and, in particular, to the Sobolev classes W 1,p loc with p ą n ´1, (see [8,12]).…”

Section: Introductionmentioning

confidence: 99%

On the asymptotic behavior at infinity of one mapping class

Klishchuk¹,

Salimov

Stefanchuk

2023

PIGC

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Section: Introductionmentioning

confidence: 99%

On the asymptotic behavior at infinity of one mapping class

Klishchuk¹,

Salimov

Stefanchuk

2023

PIGC

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“…An alternative approach to the problem of estimating the functionals in complex domains can be found in [2,3,12,14]…”

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confidence: 99%

On the Koebe Quarter Theorem for Polynomials

Стоколос

Dmitrishin

Smorodin

2022

PIGC

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The Koebe One Quarter Theorem states that the range of any Schlicht function contains the centered disc of radius 1/4 which is sharp due to the value of the Koebe function at −1. A natural question is finding polynomials that set the sharpness of the Koebe Quarter Theorem for polynomials. In particular, it was asked in [7] whether Suffridge polynomials [15] are optimal. For polynomials of degree 1 and 2 that is obviously true. It was demonstrated in [10] that Suffridge polynomials of degree 3 are not optimal and a promising alternative family of polynomials was introduced. These very polynomials were actually discovered earlier independently by M. Brandt [3] and D. Dimitrov [9]. In the current article we reintroduce these polynomials in a natural way and make a far-reaching conjecture that we verify for polynomials up to degree 6 and with computer aided proof up to degree 52. We then discuss the ensuing estimates for the value of the Koebe radius for polynomials of a specific degree.

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“…In this paper, we study the ring homeomorphisms with respect to p-modulus in R n with p > n. For these mappings we derive a lower bound for the measure of a ball image and solve an extremal problem on minimizing its volume functional. Note that the corresponding case for n = 2 has been studied in [11]- [13].…”

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confidence: 99%

An extremal problem for the volume functional

Salimov

Klishchuk

2018

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We consider the class of ring Q-homeomorphisms with respect to p-modulus in R n with p > n, and obtain a lower bound for the volume of images of a ball under such mappings. In particular, the following theorem is proved in the paper: Let D be a bounded domain in R n , n 2 and let f : D → R n be a ring Q-homeomorphism with respect to p-modulus at a point x 0 ∈ D with p > n, and the function Q satisfies the condition q x0 (t) q 0 t −α , q 0 ∈ (0, ∞) , α ∈ [0, ∞) for a.e. t ∈ (0, d 0), d 0 = dist(x 0 , ∂D). Then for all r ∈ (0, d 0) the estimate m(f B(x 0 , r)) Ω n (p − n α + p − n

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Extremal Problem for the Area of the Image of a Disk

Cited by 4 publications

References 4 publications

On the asymptotic behavior at infinity of one mapping class

On the asymptotic behavior at infinity of one mapping class

On the Koebe Quarter Theorem for Polynomials

An extremal problem for the volume functional

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