We consider the well-known problem of the geometric theory of functions of a complex variable on non-overlapping domains with free poles on radial systems. The main results of the present work strengthen and generalize several known results for this problem.
The paper deals with the following problem stated in [1] by V.N. Dubinin and earlier in different form by G.P. Bakhtina [2]. Let a 0 = 0, |a 1 | = . . . = |a n | = 1, a k ∈ B k ⊂ C, where B 0 , . . . , B n are non-overlapping domains, and B 1 , . . . , B n are symmetric domains about the unit circle. Find the exact upper bound for r γ (B 0 , 0) n k=1r(B k , a k ), where r(B k , a k ) is the inner radius of B k with respect to a k . For γ = 1 and n ≥ 2 this problem was solved by L.V. Kovalev [3,4]. In the present paper it is solved for γ n = 0, 25n 2 and n ≥ 4 under the additional assumption that the angles between neighboring line segments [0, a k ] do not exceed 2π/ √ 2γ.
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