TY - JOUR

T1 - Monotonicity and Comparison Results for Conformal Invariants

AU - Al, Baernstein II

AU - Solynin, Alexander

PY - 2013/1

Y1 - 2013/1

N2 - Abstract. Let a1, . . . , aN be points on the unit circle T with aj = eiθj ,
where 0 = θ1 ≤ θ2 ≤ · · · ≤ θN = 2π. Let Ω=C \ {a1, . . . , aN} 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{θj+1 − θj : 0 ≤ j ≤ N − 1}. How should one
choose a1, . . . , aN subject to condition δ(Ω) ≤ 2π/n so that the Poincar´e
metric λΩ(0) of Ω at the origin is as small as possible? In this paper we
answer this question by showing that λΩ(0) is minimal when Ω = Ω∗.
Several similar problems on the extremal values of the harmonic measures
and capacities are also discussed.

AB - Abstract. Let a1, . . . , aN be points on the unit circle T with aj = eiθj ,
where 0 = θ1 ≤ θ2 ≤ · · · ≤ θN = 2π. Let Ω=C \ {a1, . . . , aN} 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{θj+1 − θj : 0 ≤ j ≤ N − 1}. How should one
choose a1, . . . , aN subject to condition δ(Ω) ≤ 2π/n so that the Poincar´e
metric λΩ(0) of Ω at the origin is as small as possible? In this paper we
answer this question by showing that λΩ(0) is minimal when Ω = Ω∗.
Several similar problems on the extremal values of the harmonic measures
and capacities are also discussed.

M3 - Article

SP - 91

EP - 113

JO - Revista Matematica Iberoamericana

JF - Revista Matematica Iberoamericana

ER -