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Zeros of Differential Polynomials of Meromorphic Functions

Ta Thi Hoai An , $icon-email$ Nguyen Viet Phuong

Abstract

Let $f$ be a transcendental meromorphic function on $C, k$ be a positive integer, and $Q_{0}, Q_{1}, \dots, Q_{k}$ be polynomials in $C [z]$ . 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 $Q_{0} (f) Q_{1} (f^{'}) \dots Q_{k} (f^{(k})$ in $f$ . We will also prove that the Nevanlinna defect of the differential polynomial form $Q_{0} (f) Q_{1} (f^{'}) \dots Q_{k} (f^{(k})$ in $f$ satisfies $\sum_{a \in C} δ (a, Q_{0} (f) Q_{1} (f^{'}) \dots Q_{k} (f^{(k)})) \leq 1$ 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.

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Zeros of Differential Polynomials of Meromorphic Functions

Abstract