  We consider a monogamy inequality of quantum discord in a pure tripartite
state and show that it is equivalent to an inequality between quantum mutual
information and entanglement of formation of two parties. Since this inequality
does not hold for arbitrary bipartite states, quantum discord can generally be
both monogamous and polygamous. We also carry out numerical calculations for
some special states. The upper bounds of quantum discord and classical
correlation are also discussed and we give physical analysis on the invalidness
of a previous conjectured upper bound of quantum correlation. Our results
provide new insights for further understanding of distributions of quantum
correlations.
