From the Logarithmic Derivative Lemma to Hayman's Conjecture

The fact that the complex differential polynomial f(z)nf '(z)-a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(z)<^>{n}f'(z)-a$$\end{document} has infinitely many zeros whenever a is a non-zero constant, where f(z) is any transcendental meromorphic function and n is any positive integer. It is usually called Hayman's conjecture and has the similar versions in complex difference or q-difference polynomials. We will provide a unified proof to demonstrate that some versions are valid when f(z) is a transcendental entire function with appropriate conditions on the growth. Additionally, we consider the zeros of complex delay-differential polynomials of meromorphic functions of specific types.