Zeros of Differential Polynomials of Meromorphic Functions

Let f be a transcendental meromorphic function on \(\mathbb {C},\) k be a positive integer, and \(Q_{0},Q_{1},\dots ,Q_{k}\) be polynomials in \(\mathbb {C}[z]\). In this paper, we will prove that the frequency of distinct poles of f is governed by the frequency of zeros of the differential polynomial form \(Q_{0}(f)Q_{1}(f^{\prime }){\dots } Q_{k}(f^{(k)})\) in f. We will also prove that the Nevanlinna defect of the differential polynomial form \(Q_{0}(f)Q_{1}(f^{\prime }){\dots } Q_{k}(f^{(k)})\) in f satisfies

$$ \sum\limits_{a\in\mathbb{C}}\delta\left( a,Q_{0}(f)Q_{1}(f^{\prime}){\dots} Q_{k}\left( f^{(k)}\right)\right)\leq 1$$

with suitable conditions on k and the degree of the polynomials. Thus, our work is a generalization of Mues’s conjecture and Goldberg’s conjecture for the more general differential polynomials.