Extremal decomposition of the complex plane with free poles

A.K. Bakhtin. Scientific legacy

“…Theorem 5.4. [7] Let γ P (1; 2]. Then for any different points a 1 and a 2 of the unit circle and any pairwise non-overlapping domains B 0 , B 1 , B 2 , a 0 = 0 P B 0 Ă C, a 1 P B 1 Ă C, a 2 P B 2 Ă C, the following inequality holds…”

“…An open problem of finding the maximum of product of inner radii of two domains relative to the points of a unit circle with each power γ P (0, 2 ] of the inner radius of the domain relative to the origin, provided that all three domains are mutually non-overlapping domains is solved. It is also generalized to the case of arbitrary pair of points of the complex plane [7].…”

“…Under the O. K. Bakhtin supervision A. L. Targonskyi (2006) [12][13][14][41][42][43][44][45], V. E. Vyun (2008) [17][18][19], R. V. Podvisotskii [5,11], I. Y. Vygovska (2012) [46][47][48], I. V. Denega (2013) [7][8][9], Y. V. Zabolotnyi (2014) [20,22,26,27], L. V. Vyhivska (2019) [10,15,16], I. Y. Dvorak (2019) [10,21] defended their candidate theses.…”

Inequality for the inner radii of symmetric non-overlapping domains

“…In geometric function theory of a complex variable problems maximizing the product of inner radii of non-overlapping domains are well known [1][2][3][4][5][6][7][8][9][10]. One of the such problems is considered in the article.…”