Hyperbolic lengths and conformal embeddings of Riemann surfaces

Leth be a homeomorphic bijection between hyperbolic Riemann surfacesR andR’. If there is a conformal mapping ofR intoR’ homotopic toh, then for any hyperbolic geodesicc onR the length of the hyperbolic geodesic freely homotopic to the imageh(c) is less than or equal to the hyperbolic length ofc. We show that the converse is not necessarily true.