  Let F and G be two families of meromorphic functions on a domain D, and let
a, b and c be three distinct points in the extended complex plane. Let G be a
normal family in D such that all limit functions of G are non-constant. If for
each f in F, there exists g in G such that f and g share a, b and c partially,
then F is normal in D. This gives a sharp improvement of a result due to X. J.
Liu, S. H. Li and X. C. Pang. We also prove some interesting related sharp
results.
