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Picard-Hayman behavior of derivatives of meromorphic functions
Comptes Rendus. Mathématique, Volume 358 (2020) no. 6, pp. 753-756.

Let f be a transcendental meromorphic function on , and P(z),Q(z) be two polynomials with degP(z)>degQ(z). In this paper, we prove that: if f(z)=0f (z)=a(a nonzero constant), except possibly finitely many, then f (z)-P(z)/Q(z) has infinitely many zeros. Our result extends or improves some previous related results due to Bergweiler–Pang, Pang–Nevo–Zalcman, Wang–Fang, and the author, et. al.

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DOI : 10.5802/crmath.96
Classification : 30D35, 30D45
Yan Xu 1 ; Shirong Chen 1 ; Peiyan Niu 2

1 School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, P.R.China
2 School of Mathematics and Statistics, Changshu Institute of Technology, Changshu 215500, P.R.China
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {Picard-Hayman behavior of derivatives of meromorphic functions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {753--756},
     publisher = {Acad\'emie des sciences, Paris},
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     year = {2020},
     doi = {10.5802/crmath.96},
     language = {en},
}
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Yan Xu; Shirong Chen; Peiyan Niu. Picard-Hayman behavior of derivatives of meromorphic functions. Comptes Rendus. Mathématique, Volume 358 (2020) no. 6, pp. 753-756. doi : 10.5802/crmath.96. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.96/

[1] Walter Bergweiler; Xuecheng Pang On the derivatives of meromorphic functions with multiple zeros, J. Math. Anal. Appl., Volume 278 (2003) no. 2, pp. 285-292 | DOI | MR | Zbl

[2] Walter K. Hayman Picard values of meromorphic functions and their derivatives, Ann. Math., Volume 70 (1959), pp. 9-42 | DOI | MR | Zbl

[3] Walter K. Hayman Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, 1964 | Zbl

[4] Olli Lehto; K. I. Virtanen On the behaviour of meromorphic functions in the neighbourhood of an isolated singularity, Ann. Acad. Sci. Fenn., Ser. A I, Volume 240 (1957), pp. 1-9 | MR | Zbl

[5] Shahar Nevo; Xuecheng Pang; Lawrence Zalcman Picard–Hayman behavior of derivatives of meromorphic functions with multiple zeros, Electron. Res. Announc. Am. Math. Soc., Volume 12 (2006), pp. 37-43 | DOI | MR | Zbl

[6] Xuecheng Pang; Shahar Nevo; Lawrence Zalcman Derivatives of meromorphic functions with multiple zeros and rational functions, Comput. Methods Funct. Theory, Volume 8 (2008) no. 2, pp. 483-491 | DOI | MR | Zbl

[7] Xuecheng Pang; Lawrence Zalcman Normal families and shared values, Bull. Lond. Math. Soc., Volume 32 (2000) no. 3, pp. 325-331 | DOI | MR | Zbl

[8] Yuefei Wang; Mingliang Fang Picard values and normal families of meromorphic functions with multiple zeros, Acta Math. Sin., New Ser., Volume 14 (1998) no. 1, pp. 17-26 | DOI | MR | Zbl

[9] Yan Xu Picard values and derivatives of meromorphic functions, Kodai Math. J., Volume 28 (2005) no. 1, pp. 99-105 | MR | Zbl

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