Normal families of meromorphic functions and shared functions

Abstract

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 (≠≡ 0,∞) 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 ∈ F, at each common zero of f and h the multiplicities mf for f and mh for h satisfy mf ≥ mh + k + 1 for k > 1 and mf ≥ 2mh + 3 for k = 1, and at each common pole of f and h, the multiplicities nf for f and nh for h satisfy nf ≥ nh + 1, then the family F is normal on D.