Abstract
We say that two meromorphic functions f and g share a small function \(\alpha \) counting multiplicities if \(f-\alpha \) and \(g-\alpha \) admit the same zeros with the same multiplicities. Let Q be a polynomial of one variable. In [Comput. Methods Funct. Theory 17: 613-634, 2017, Theorem. 1.1], we proved that if \((Q(f))^{(k)}\) and \((Q(g))^{(k)}\) share \(\alpha \) counting multiplicities then, with suitable conditions on the degree of Q and on the number of zeros and the multiplicities of the zeros of \(Q'\), there are explicit relations between Q(f) and Q(g). Unfortunately, there is a gap at the beginning of the proof of An and Phuong (Comput. Methods Funct. Theory 17:613–634, 2017, Theorem 1.1]. We will give a way to avoid the gap. This proof can also be used to fix the gaps of other authors’ published papers listed in Schweizer (arXiv:1705.05048v2, 2017).
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Acknowledgements
The authors would like to thank Andreas Schweizer for informing us of the error in the proof of [1, Thm. 1.1] and again for many useful discussions. We would like to thank the referees for carefully reading our manuscript and for giving such constructive comments which substantially helped to improve the clarity of the paper. A part of this article was written while the first author was visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM). She would like to thank the institute for its warm hospitality and partial support.
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The authors are supported by Vietnam’s National Foundation for Science and Technology Development (NAFOSTED) and by the International Center for Research and Postgraduate Training in Mathematics (ICRTM)
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An, T.T.H., Phuong, N.V. A Lemma about Meromorphic Functions Sharing a Small Function. Comput. Methods Funct. Theory 22, 277–286 (2022). https://doi.org/10.1007/s40315-021-00388-3
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DOI: https://doi.org/10.1007/s40315-021-00388-3