### Abstract

Let a_{1}, . . . , aN be points on the unit circle T with a _{j} = e^{iθj} , where 0 = θ_{1} ≤ θ_{2} ≤ · · · ≤ θ_{N} = 2π. Let Ω = C \ {a_{1}, . . . , 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 a_{1}, . . . , 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.

Original language | English |
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Pages (from-to) | 91-113 |

Number of pages | 23 |

Journal | Revista Matematica Iberoamericana |

Volume | 29 |

Issue number | 1 |

DOIs | |

State | Published - 2013 |

### Keywords

- Capacity
- Comparison theorem
- Harmonic measure
- Hyperbolic metric

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## Cite this

Baernstein, A., & Solynin, A. Y. (2013). Monotonicity and comparison results for conformal invariants.

*Revista Matematica Iberoamericana*,*29*(1), 91-113. https://doi.org/10.4171/rmi/714