Relative primeness of entire functions (Q1281595)

Recently, factorization theory for transcendental functions has received the attention of some mathematicians working in value distribution theory or, more generally, complex analysis. For some nonconstant entire function \(F\) a decomposition \(F=f \circ g\) is called factorization of \(F\) with \(f\) and \(g\) being the left and right factor, resp. Here \(f\) is assumed to be meromorphic and \(g\) to be entire. The greatest common right factor (GCRF) of two given functions \(F\) and \(G\) is defined as the entire function \(h\) such that \(h_1\) is a right factor of \(h\) whenever \(h_1\) is a right factor of both, \(F\) and \(G\). Consequently, \(F\) and \(G\) are called relatively prime if their GCRF is a linear function. The main result of the present paper is Theorem. Let \(F\) and \(G\) be entire transcendental functions such that \(FG\) is prime. Then \(F\) and \(G\) are relatively prime unless \(F = f \circ h\) and \(G = g \circ h\) where \(h\) is a nonlinear prime entire function and \(f\) and \(g\) satisfy one of the following cases: (i) \(f = Le^\alpha\) and \(g=e^{-\alpha}\), (ii) \(g = e^\alpha\) and \(g=Le^{-\alpha}\), with some entire function \(\alpha\) and some linear \(L\). The proof bases on and follows C. C. Yang's proof of a slightly weaker result: Theorem (C. C. Yang). Let \(F\) and \(G\) be transcendental entire functions such that \(FG\) is prime and satisfying the growth condition \(T(r,F)=S(r,g)\) or \(T(r,G)=S(r,F)\) Then \(F\) and \(G\) are relatively prime.