Hyperbolic lengths of geodesics surrounding two punctures

Abstract:

For the plane regions ${\Omega _1} = \left \{ {\left | z \right | < R,z \ne 0,1} \right \}$ with $R > 1$, and ${\Omega _2} = {\mathbf {C}} \setminus \left \{ {0,1,p} \right \}$ with $\left | p \right | = R > 1$, we describe, as $R \to \infty$, the hyperbolic lengths of the geodesies surrounding 0 and 1. Upper and lower bounds for the lengths are also stated, and these results are used to obtain inequalities, which are precise in a certain sense, for the length of the geodesic surrounding 0 and 1 in an arbitrary plane region $\Omega$ satisfying ${\Omega _1} \subset \Omega \subset {\Omega _2}$.