Abstract
In this paper, we studied a normality criterion concerning Hayman’s question and proved: let \(n({\ge }2),k({\ge }1), m({\ge }0)\) be three integers, let \(h(z)({\not \equiv } 0)\) be a holomorphic function in a domain D with all zeros that have multiplicity at most m, and let \(\mathcal {F}\) be a family of functions meromorphic in a domain D, all of whose zeros have multiplicity at least \(k+m\). If, for any two functions \(f, g\in \mathcal {F}\), \(f^nf^{(k)}\) and \(g^ng^{(k)}\) share h(z) in D, then \(\mathcal {F}\) is normal in D. The result gets rid of two conditions “all zeros of h(z) have multiplicity divisible by \(n+1\)” and “all poles of f(z) have multiplicity at least \(m+1\)” in the result due to Meng and Hu (Bull Malays Math Sci Soc 38:1331–1347, 2015).
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The authors would like to thank the referee for a careful reading of the manuscript and some valuable suggestions.
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Communicated by See Keong Lee.
Research supported by the NNSF of China (Grant No. 11371149) and the Graduate Student Overseas Study Program from South China Agricultural University (Grant No. 2017LHPY003).
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Deng, B., Lei, C. & Fang, M. Normal Families and Shared Functions Concerning Hayman’s Question. Bull. Malays. Math. Sci. Soc. 42, 847–857 (2019). https://doi.org/10.1007/s40840-017-0515-7
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DOI: https://doi.org/10.1007/s40840-017-0515-7