Estimates of the Carath\'eodory metric on the symmetrized polydisc
  Estimates for the Carath\'eodory metric on the symmetrized polydisc are obtained. It is also shown that the Carath\'eodory and Kobayashi distances of the symmetrized three-disc do not coincide. 
N. Nikolov, P. Pflug, P. J. Thomas and W. Zwonek