A note on Hayman's problem

TL;DR

This paper proves that certain differential polynomials of transcendental meromorphic functions have infinitely many zeros and fixed points, addressing open problems in complex analysis and zero distribution theory.

Contribution

It establishes the infinite zeros of differential polynomials and fixed points of derivatives, extending results to partial differential polynomials with small coefficients.

Findings

Differential polynomial $Q(f)^{(k)}-p$ has infinitely many zeros.

$Q(f)^{(k)}$ has infinitely many fixed points.

Provides a unified approach to zero distribution of differential polynomials.

Abstract

In this note, it is shown that the differential polynomial of the form $Q(f)^{(k)}-p$ has infinitely many zeros, and particularly $Q(f)^{(k)}$ has infinitely many fixed points for any positive integer $k$, where $f$ is a transcendental meromorphic function, $p$ is a nonzero polynomial and $Q$ is a polynomial with coefficients in the field of small functions of $f$. The results are traced back to Problem 1.19 and Problem 1.20 in the book of research problems by Hayman and Lingham. As a consequence, we give an affirmative answer to an extended problem on the zero distribution of $(f^n)'-p$, proposed by Chiang and considered by Bergweiler. Moreover, our methods provide a unified way to study the problem of the zero distributions of partial differential polynomials of meromorphic functions in one and several complex variables with small meromorphic coefficients.