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 f and g g are nonlinear entire functions such that f ( g ) f(g) is transcendental and of finite order, then f ( g ) f(g) has infinitely many fix-points. (2) If f f is a polynomial of degree ≧ 3 \geqq 3 , and g g is an arbitrary transcendental meromorphic function, then f ( g ) f(g) must have infinitely many fix-points. (3) Let p ( z ) , q ( z ) p(z),q(z) be any nonconstant polynomials, at least one of which is not c c -even, and let a a and b b be any constants with a a or b ≠ 0 b \ne 0 . Then h ( z ) = q ( z ) exp ⁡ ( a z 2 + b z ) + p ( z ) h(z) = q(z)\exp (a{z^2} + bz) + p(z) is prime.