TY - JOUR
T1 - Monotonicity and comparison results for conformal invariants
AU - Baernstein, Albert
AU - Solynin, Alexander Yu
PY - 2013
Y1 - 2013
N2 - Let a1, . . . , aN be points on the unit circle T with a j = 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 the condition δ(Ω) ≤ 2π/n so that the Poincaré metric λΩ(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.
AB - Let a1, . . . , aN be points on the unit circle T with a j = 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 the condition δ(Ω) ≤ 2π/n so that the Poincaré metric λΩ(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.
KW - Capacity
KW - Comparison theorem
KW - Harmonic measure
KW - Hyperbolic metric
UR - http://www.scopus.com/inward/record.url?scp=84876809619&partnerID=8YFLogxK
U2 - 10.4171/rmi/714
DO - 10.4171/rmi/714
M3 - Article
AN - SCOPUS:84876809619
VL - 29
SP - 91
EP - 113
JO - Revista Matematica Iberoamericana
JF - Revista Matematica Iberoamericana
SN - 0213-2230
IS - 1
ER -