Normal families of meromorphic functions and shared functions

The paper is devoted to the normal families of meromorphic functions and shared functions. Generalizing a result of Chang (2013), we prove the following theorem. Let h (not equal a parts per thousand 0,a) be a meromorphic function on a domain D and let k be a positive integer. Let F be a family of meromorphic functions on D, all of whose zeros have multiplicity at least k + 2, such that for each pair of functions f and g from F, f and g share the value 0, and f ((k)) and g ((k)) share the function h. If for every f a F, at each common zero of f and h the multiplicities m (f) for f and m (h) for h satisfy m (f) a parts per thousand yen m (h) + k + 1 for k > 1 and m (f) a parts per thousand yen 2m (h) + 3 for k = 1, and at each common pole of f and h, the multiplicities nf for f and nh for h satisfy n (f) a parts per thousand yen n (h) + 1, then the family F is normal on D.