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On unavoidable families of meromorphic functions

Part of: Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  18 November 2021

Thierry Meyrath
Thierry Meyrath*
Affiliation:
Department of Mathematics, University of Luxembourg, 6, Avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg
*
e-mail: thierry.meyrath@uni.lu
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Abstract

We prove several results on unavoidable families of meromorphic functions. For instance, we give new examples of families of cardinality 3 that are unavoidable with respect to the set of meromorphic functions on $\mathbb C$ . We further obtain families consisting of less than three functions that are unavoidable with respect to certain subsets of meromorphic functions. In the other direction, we show that for every meromorphic function f, there exists an entire function that avoids f on $\mathbb C$ .

Keywords

Unavoidable familieszero–one setsNevanlinna Theory

MSC classification

Primary: 30D30: Meromorphic functions, general theory 30D35: Distribution of values, Nevanlinna theory 30D20: Entire functions, general theory
Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 1 , March 2023 , pp. 114 - 123
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Copyright
© Canadian Mathematical Society, 2021

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References

Bergweiler, W. and Eremenko, A., Radially distributed values and normal families. Int. Math. Res. Not. (IMRN). 23(2019), 7356–7378.CrossRefGoogle Scholar
Bergweiler, W., Eremenko, A., and Hinkkanen, A., Entire functions with two radially distributed values. Math. Proc. Camb. Phil. Soc. 165(2018), 93–108.CrossRefGoogle Scholar
Gol’dberg, A. A. and Ostrovskii, I. V., Value distribution of meromorphic functions, Translations of Mathematical Monographs, 236, American Mathematical Society, Providence, RI, 2008.CrossRefGoogle Scholar
Hayman, W. K., Meromorphic functions, Clarendon Press, Oxford, 1964.Google Scholar
Hayman, W. K. and Rubel, L. A., Unavoidable systems of functions. Math. Proc. Camb. Phil. Soc. 117(1995), 345–351.CrossRefGoogle Scholar
Lappan, P., Avoidance criteria for normal families and normal functions, In: Begehr, H. G. W., Gilbert, R. P., and Wong, M. W. (eds.), Progress in analysis: Proceedings of the 3rd International ISAAC Congress. Vol. 1, World Scientific, Singapore, 2003, pp. 221–228.CrossRefGoogle Scholar
Lappan, P., Special classes of normal families. Comput. Methods Funct. Theory. 8(2008), 133–142.CrossRefGoogle Scholar
Lappan, P., A continuous function that no meromorphic function can avoid. Ann. Acad. Sci. Fenn. Math. 34(2009), 173–177.Google Scholar
Logvinenko, V. N., On rational functions with disjoint graphs. Complex Var. Elliptic Equ. 26(1995), 255–271.Google Scholar
Ozawa, M., On the zero-one set of an entire function. Kodai Math. Sem. Rep. 28(1977), 311–316.CrossRefGoogle Scholar
Ozawa, M., On the zero-one set of an entire function II. Kodai Math. J. 2(1979), 194–199.CrossRefGoogle Scholar
Rubel, L. A. and Yang, C. C., Interpolation and unavoidable families of meromorphic functions. Michigan Math. J. 20(1974), 289–296.CrossRefGoogle Scholar
Rudin, W., Real and complex analysis. 3rd ed., McGraw Hill, New York, 1987.Google Scholar
Winkler, J., Zur Existenz ganzer Funktionen bei vorgegebener Menge der Nullstellen und Einsstellen. Math. Z. 168(1979), 77–85.CrossRefGoogle Scholar
Yang, L., Value distribution theory, Springer, Berlin, Heidelberg, 1993.Google Scholar