Normal families of meromorphic functions with multiple values

Let F be a family of meromorphic functions defined in a domain D, let ψ ( ≢ 0 ) be a holomorphic function in D, and k be a positive integer. Suppose that, for every function f ∈ F , f ≠ 0 , f ( k ) ≠ 0 , and all zeros of f ( k ) − ψ ( z ) have multiplicities at least ( k + 2 ) / k . If, for k = 1 , ψ has only zeros with multiplicities at most 2, and for k ⩾ 2 , ψ has only simple zeros, then F is normal in D. This improves and generalizes the related results of Gu, Fang and Chang, Yang, Schwick, et al.