  We analyze a recently found inequality for eigenvalues of the density matrix
and purity parameter describing either a bipartite system state or a single
qudit state. The Minkowski type trace inequality for the density matrices of
the qudit states is rewritten in terms of the purities. The properties of the
obtained inequality are discussed. The $X$-states of the two qubits and the
single qudit are considered in detail. A study of the relation of the obtained
purity inequalities with the entanglement is presented.
