On some results of Mues, Bergweiler and Eremenko concerning differential polynomials. (English) Zbl 07925648
Summary: In this paper, we shall prove the following result: Let \(f\) be a transcendental meromorphic function, and let \(P(z)\) be a polynomial with \(\deg P\geq 2\). Then \([P(f(z))]^{(k)}\) takes every non-zero complex value infinitely often for \(k = 1, 2, 3, \ldots\), by making use of, among other things, particularly an important result of
K. Yamanoi [Proc. Lond. Math. Soc. (3) 106, No. 4, 703–780 (2013; Zbl 1300.30067)]. This improves the results due to E. Mues [Arch. Math. 32, 55–67 (1979; Zbl 0407.30022)], W. Bergweiler and A. Eremenko [Rev. Mat. Iberoam. 11, No. 2, 355–373 (1995; Zbl 0830.30016)], etc. Moreover, the corresponding normality criterion is also obtained.
MSC:
| 30D30 | Meromorphic functions of one complex variable (general theory) |
| 30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |
| 30D45 | Normal functions of one complex variable, normal families |
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