Harmonic measure of radial line segments and symmetrization

Let l_k = {z : arg z = α_k, r₁ ≤ |z\ ≤ r₂} for k = 1, . . ., n, 0 < r₁ < r₂ ≤ 1, and α_k ∈ ℝ; let E = Uⁿ_k=1 l_k; let E* = {z : arg zⁿ = 0, r₁ ≤ |z| ≤ r₂}; and let ω_E(z) be the harmonic measure of E with respect to the domain {z : \z\ <1} \ E. The inequality ω_E(0) ≤ ω_E* (0), is established, which solves the problem of Gonchar on the harmonic measure of radial slits. The proof uses the dissymmetrization method of Dubinin and the method of the extremal metric in the form of the problem of extremal partitioning into non-overlapping domains.