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Normal Families and Shared Functions

Abstract Let k ∈ ℕ, m ∈ℕ ∪{0}, and let a(z)(≢ 0) be a holomorphic function, all zeros of a(z) have multiplicities at most m. Let ℱ be a family of meromorphic functions in D. If for each f ∈ℱ, the zeros of f have multiplicities at least k + m + 1 and all poles of f are of multiplicity at least m + 1, and for f,g ∈ℱ, ff(k)−a(z) and gg(k)−a(z) share 0, then ℱ is normal in D. Some examples are given to show that the conditions are best, and the result removes the condition “m is an even integer” in the result due to Sun [Kragujevac Journal of Math 38(2), 173-282, 2014].