<abstract xmlns="http://www.w3.org/1999/xhtml">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<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_auom-2019-0004_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>r</mi></mrow><mi>γ</mi></msup><mo>(</mo><msub><mrow><mi>B</mi></mrow><mn>0</mn></msub><mo>,</mo><mn>0</mn><mo>)</mo><munderover><mo>∏</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>r</mi><mo>(</mo><msub><mrow><mi>B</mi></mrow><mi>k</mi></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mi>k</mi></msub><mo>)</mo><mo>,</mo></mrow></math><tex-math>$${r^\gamma }({B_0},0)\prod\limits_{k = 1}^n r ({B_k},{a_k}),$$</tex-math></alternatives></disp-formula>where <italic>B</italic>0, <italic>B</italic>1, <italic>B</italic>2,..., <italic>Bn</italic>, <italic>n</italic> ≥ 2, are pairwise disjoint domains in ¯𝔺, <italic>a</italic>0 = 0, |<italic>ak</italic>| = 1, <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_auom-2019-0004_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>k</mi><mo>=</mo><mover><mrow><mn>1</mn><mo>,</mo><mi>n</mi></mrow><mo>¯</mo></mover></mrow></math><tex-math>$k = \overline {1,n}$</tex-math></alternatives></inline-formula>and <italic>γ</italic> ∈ 2 (0; <italic>n</italic>], <italic>r</italic>(<italic>B</italic>, <italic>a</italic>) is the inner radius of the domain, <italic>B</italic> ⊂ ¯𝔺, with respect to a point <italic>a</italic> ∈ <italic>B</italic>. In the paper we consider a more general problem in which restrictions on the geometry of the location of points <italic>ak</italic>, <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_auom-2019-0004_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>k</mi><mo>=</mo><mover><mrow><mn>1</mn><mo>,</mo><mi>n</mi></mrow><mo>¯</mo></mover></mrow></math><tex-math>$k = \overline {1,n}$</tex-math></alternatives></inline-formula>are weakened.</abstract>

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 functionalrγ(B0,0)∏k=1nr(Bk,ak),$${r^\gamma }({B_0},0)\prod\limits_{k = 1}^n r ({B_k},{a_k}),$$where B0, B1, B2,..., Bn, n ≥ 2, are pairwise disjoint domains in ¯𝔺, a0 = 0, |ak| = 1, k=1,n¯$k = \overline {1,n}$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=1,n¯$k = \overline {1,n}$are weakened.

Problem on extremal decomposition of the complex plane

Department of complex analysis and potential theory, Institute of mathematics of the National Academy of Sciences of Ukraine

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Problem on extremal decomposition of the complex plane"}

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...

Problem on extremal decomposition of the complex plane

Published Online: Mar 02, 2019

Page range: 61 - 77

Accepted: Mar 30, 2018

DOI: https://doi.org/10.2478/auom-2019-0004

Keywords
Inner radius of the domain, non-overlapping domains, -radial system of points, separating transformation, quadratic differential, Green’s function

© 2019 Iryna Denega et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Problem on extremal decomposition of the complex plane

Published Online: Mar 02, 2019

Page range: 61 - 77

Accepted: Mar 30, 2018

DOI: https://doi.org/10.2478/auom-2019-0004

KeywordsInner radius of the domain, non-overlapping domains, -radial system of points, separating transformation, quadratic differential, Green’s function

© 2019 Iryna Denega et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Inner radius of the domain, non-overlapping domains, -radial system of points, separating transformation, quadratic differential, Green’s function