Problem on extremal decomposition of the complex plane

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 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.

A.K. Bakhtin. Scientific legacy

“…Theorem 5.3. [27] For any natural n ě e 9 and 0 ă γ ď n 2 3 ´1 ? ln n the following inequality holds…”

“…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.…”