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Normal families and shared function II

Chengxiong Sun

Fasciculi Mathematici

2019

nr 62

141--154

Let k, n ∈ N,l ∈ N\ {1} , m ∈ N U {0}, and a(z)(≠ 0) be a holomorphic function, all of whose zeros have multiplicities at most m. Let F be a family of meromorphic functions in D such that multiplicities of zeros of each f ∈ F are at least k + m. If for f, g ∈ F satisfy fl(f(k))n and gl(g(k))n share a(z), then F is normal in D. The examples are provided to show that the result is sharp. The result extends the related theorems [9,10,12]. we also omit the conditions “m is divisible by n +1” and “all poles of f have multiplicities at least m + 1” in the result due to Meng, Liu and Xu [12] [Journal of Computational Analysis and Applications 27(3)(2019), 511-526].