Some results on primeness and unique factorization of entire functions (Q1804780)

A function \(F\), meromorphic and transcendental in \(\mathbb{C}\), is called factorizable, if it can be written as \(F(z)= f(g(z))\), where \(f\) is transcendental meromorphic and \(g\) is nonlinear entire, or \(f\) is nonlinear rational and \(g\) is transcendental meromorphic. It is called uniquely factorizable, if, for any other factorization \(F(z)= f_1(g_1(z))\), \(f_1(z)= f(L(z))\) and \(g_1(z)= L^{- 1}(g(z))\) hold, \(L\) some Möbius transform. \(F\) is called prime, if no nontrivial factorization exists. The paper under review contains several criteria for primeness and unique factorizibility, e.g.: Theorem 1: \[ F(z)= z+ k_1(e^z)+ k_2(e^{P_1(e^z)})+ k_3(e^{P_2(e^{P_3(e^z)})}) \] is prime, if \(P_1\), \(P_2\), \(P_3\), \(k_3\) are non-constant polynomials, \(k_2(e^z)\) has finite order and if order of \(k_1< \) degree of \(P_1\). Theorem 2: \[ F(z)= (ze^{P(z)})\circ (ze^{Q(e^{R(e^z)})}) \] and \[ F(z)= (z+ P(e^z))\circ (z+ Q(e^{R(e^z)})) \] are unique factorizable (\(P\), \(Q\), \(R\) polynomials). The proofs use standard methods of Nevanlinna theory.