The fact that the complex differential polynomial \(f(z)^{n}f'(z)-a\) 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.
