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Quasinormality and exceptional functions of derivatives

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Abstract

Let k be a positive integer, \(\{f_{n}(z)\}\) be a family of functions meromorphic on a domain D, whose zeros all have multiplicity at least k + 2, and let \(h(z)(\not \equiv 0)\) be a function holomorphic on D. Suppose \(f^{(k)}_{n}(z)\ne h(z)\) on D, and no subsequence of \(\{f_{n}\}\) is normal at some \(z_{0}\in D\). In this paper, we prove that the limit function \(f^{(k)}(z)\) of \(\{f^{(k)}_{n}(z)\}\) is equal to the exceptional function h(z) of derivatives \(f^{(k)}_{n}(z)\), and some characteristics of the subsequence of \(\{f_{n}(z)\}\) are obtained near the nor-normal point \(z_0\). As applications of this result, some quasinormal criteria are also given.

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Acknowledgements

We thank the referee for his/her valuable comments and suggestions made to this paper.

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Correspondence to Yan Xu.

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Research supported by NNSF of China (Grant No. 11471163).

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Hu, Y., Chen, S. & Xu, Y. Quasinormality and exceptional functions of derivatives. Anal.Math.Phys. 11, 174 (2021). https://doi.org/10.1007/s13324-021-00610-4

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