A note on Hayman's conjecture

In this paper, we will give suitable conditions on differential polynomials Q(f) such that they take every finite nonzero value infinitely often, where f is a meromorphic function in complex plane. These results are related to Problems 1.19 and 1.20 in a book of Hayman and Lingham [Research Problems in Function Theory, preprint (2018), https://arxiv.org/pdf/1809.07200.pdf]. As consequences, we give a new proof of the Hayman conjecture. Moreover, our results allow differential polynomials Q(f) to have some terms of any degree of f and also the hypothesis n > k in [Theorem 2 of W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana 11(2) (1995) 355-3731 is replaced by n >= 2 in our result.