Picard-Hayman behavior of derivatives of meromorphic functions

Abstract

Let f be a transcendental meromorphic function on C, and P (z),Q(z) be two polynomials with degP (z) > degQ(z). In this paper, we prove that: if f (z) = 0 ⇒ f ′(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. 2020 Mathematics Subject Classification. 30D35, 30D45. Funding. This work was supported by NSFC(Grant No.11471163). Manuscript received 7th January 2020, accepted 4th July 2020.