Inequalities for the internal radii of non-overlapping domains

Bakhtin, A. K.; Vygivska, Liudmyla V.; Denega, Iryna

doi:10.1007/s10958-016-3201-7

Cited by 11 publications

(4 citation statements)

References 4 publications

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“…So, we have to investigate the case α 0 √ γ < 2. Using the results of the papers [19,26,27], we obtain the validity of the Theorem 1 and for this case too.…”

mentioning

confidence: 68%

Problem on extremal decomposition of the complex plane

Denega

Zabolotnii

2019

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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In geometric function theory of a complex variable problems on extremal decomposition with free poles on the unit circle are well known. One of such problem is the problem on maximum of the functional $${r^\gamma }({B_0},0)\prod\limits_{k = 1}^n r ({B_k},{a_k}),$$ where B0, B1, B2,..., Bn, n ≥ 2, are pairwise disjoint domains in ¯𝔺, a0 = 0, |ak| = 1, $k = \overline {1,n}$ and γ ∈ 2 (0; n], r(B, a) is the inner radius of the domain, B ⊂ ¯𝔺, with respect to a point a ∈ B. In the paper we consider a more general problem in which restrictions on the geometry of the location of points ak, $k = \overline {1,n}$ are weakened.

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“…So, we have to investigate the case α 0 √ γ < 2. Using the results of the papers [19,26,27], we obtain the validity of the Theorem 1 and for this case too.…”

mentioning

confidence: 68%

Problem on extremal decomposition of the complex plane

Denega

Zabolotnii

2019

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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“…Nowadays, this task is not completely solved, its solutions for are only known certain particular cases (see, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]). In [1,2] the problem was solved for any n 2 and γ = 1.…”

Section: Let Us Consider An Open Extremal Problem Which Was Formulatementioning

confidence: 99%

“…In the geometric function theory of complex variable extremal problems on non-overlapping domains are well-known classic direction. A lot of such problems are reduced to determination of the maximum of the product of inner radii on the system of non-overlapping domains satisfying certain conditions (see, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]). Let C be the complex plane, C = C ∪ {∞} be a point compactification, N, R be the sets of natural and real numbers, respectively, and R + = (0, ∞).…”

mentioning

confidence: 99%

Weakened problem on extremal decomposition of the complex plane

Bakhtin¹,

Denega²,

Shunkin³

2019

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The paper deals with the problem of the maximum of the functional r γ (B 0 , 0) n ∏ k=1 r (B k , a k) , where B 0 ,..., B n , n 2, are pairwise disjoint domains in C, a 0 = 0, |a k | = 1, k ∈ {1,. .. , n} and γ ∈ (0, n] (r(B, a) is the inner radius of the domain B ⊂ C with respect to a). We show that the functional attains its maximum at a configuration of the domains B k and the points a k possessing rotational n-symmetry. The proof is due to Dubinin [1] for γ = 1 and to Kuz'mina [3] for 0 < γ < 1. Subsequently, Kovalev [4] solved this problem for n 5 under the additional assumption that the angles between neighbouring line segments [0, a k ] do not exceed 2π/ √ γ.

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“…Note, that in [21] the problem was solved for n 4 and 0 < γ n but for condition α 0 2 √ γ (see also, [22,25]). Thus taking Lemma 1 into account we consider the case when…”

Section: γ) Then the Following Inequality Holdsmentioning

confidence: 99%