  In this paper we continue our earlier investigations on normal families of
meromorphic functions\cite{CS2}. Here, we prove some value distribution results
which lead to some normality criteria for a family of meromorphic functions
involving the sharing of a holomorphic function by more general differential
polynomials generated by members of the family and get some recently known
results extended and improved. In particular, the main result of this paper
leads to a counterexample to the converse of Bloch's principle.
