Further Results on Prime Entire Functions

Let H denote the set of all the entire functions $f(z)$ of the form: $f(z) \equiv h(z){e^{p(z)}} + k(z)$ where $p(z)$ is a nonconstant polynomial of degree m, and $h(\nequiv \;0)$, $k(\nequiv$ constant) are two entire functions of order less than m. In this paper, a necessary and sufficient condition for a function in H to be a prime is established. Several generalizations of known results follow. Some sufficient conditions for primeness of various subclasses of H are derived. The methods used in the proofs are based on Nevanlinna’s theory of meromorphic functions and some elementary facts about algebraic functions.