Let be points on the unit circle with , where . Let and let be with the -th roots of unity removed. The maximal gap of is defined by . How should one choose subject to the condition so that the Poincaré metric of at the origin is as small as possible? In this paper we answer this question by showing that is minimal when . Several similar problems on the extremal values of the harmonic measures and capacities are also discussed.
Cite this article
Albert Baernstein II, Alexander Yu. Solynin, Monotonicity and comparison results for conformal invariants. Rev. Mat. Iberoam. 29 (2013), no. 1, pp. 91–113DOI 10.4171/RMI/714