The fix-points and factorization of meromorphic functions

Abstract:

In this paper, we use the Nevanlinna theory of meromorphic functions and a result of Goldstein to generalize some known results in factorization and fixpoints of entire functions. Specifically, we prove (1) If $f$ and $g$ are nonlinear entire functions such that $f(g)$ is transcendental and of finite order, then $f(g)$ has infinitely many fix-points. (2) If $f$ is a polynomial of degree $\geqq 3$, and $g$ is an arbitrary transcendental meromorphic function, then $f(g)$ must have infinitely many fix-points. (3) Let $p(z),q(z)$ be any nonconstant polynomials, at least one of which is not $c$-even, and let $a$ and $b$ be any constants with $a$ or $b \ne 0$. Then $h(z) = q(z)\exp (a{z^2} + bz) + p(z)$ is prime.