Monotonicity and comparison results for conformal invariants

Let a₁, . . . , aN be points on the unit circle T with a _j = e^iθj , where 0 = θ₁ ≤ θ₂ ≤ · · · ≤ θ_N = 2π. Let Ω = C \ {a₁, . . . , a_N} and let Ω* be C with the n-th roots of unity removed. The maximal gap δ(Ω) of Ω is defined by δ(Ω) = max{θ_j+1 - θ_j: 0 ≤ j ≤ N - 1}. How should one choose a₁, . . . , a_N subject to the condition δ(Ω) ≤ 2π/n so that the Poincaré metric λ_Ω(0) of Ω at the origin is as small as possible? In this paper we answer this question by showing that λ _Ω(0) is minimal when Ω = Ω*. Several similar problems on the extremal values of the harmonic measures and capacities are also discussed.