  Relations between Shannon entropy and Renyi entropies of integer order are
discussed. For any N-point discrete probability distribution for which the
Renyi entropies of order two and three are known, we provide an lower and an
upper bound for the Shannon entropy. The average of both bounds provide an
explicit extrapolation for this quantity. These results imply relations between
the von Neumann entropy of a mixed quantum state, its linear entropy and
traces.
