Hyperbolic lengths and conformal embeddings of Riemann surfaces

Let h be a homeomorphic bijection between hyperbolic Riemann surfaces R and R'. If there is a conformal mapping of R into R' homotopic to h, then for any hyperbolic geodesic c on R the length of the hyperbolic geodesic freely homotopic to the image h(c) is less than or equal to the hyperbolic length of c. We show that the converse is not necessarily true.