On Some Results of Mues, Bergweiler and Eremenko Concerning Differential Polynomials

In this paper, we shall prove the following result: Let f be a transcendental meromorphic function, and let P(z) be a polynomial with \(\deg P\ge 2\) $degP\ge 2$ . Then \([P(f(z))]^{(k)}\) ${[P(f(z))]}^{(k)}$ takes every non-zero complex value infinitely often for \(k = 1, 2, 3, \ldots \) $k=1,2,3,\dots$ , by making use of, among other things, particularly an important result of Yamanoi (Proc Lond Math Soc 106:703–780, 2013). This improves the results due to Mues (Arch Math 32:55–67, 1979), Bergweiler and Eremenko (Rev Mat Iber 22:355–373, 1995), etc. Moreover, the corresponding normality criterion is also obtained.