Intrinsic metrics in ring domains

Three hyperbolic type metrics including the triangular ratio metric, the $j^*$-metric and the M\"obius metric are studied in an annular ring. The Euclidean midpoint rotation is introduced as a method to create upper and lower bounds for these metrics, and their sharp inequalities are found. A new M\"obius-invariant lower bound is proved for the conformal capacity of a general ring domain by using a symmetric quantity defined with the M\"obius metric.