Permutability of entire functions (Q1209432)

The authors prove a number of results on permutability of entire functions. Their main result follows: Theorem. Let \(f(z)=\text{sin } z+Q(z)\), where \(Q(z)\) is a polynomial and \(g(z)\) is a nonlinear entire function of finite order, which is permutable with \(f(z)\). Then (i) when \(\deg Q=0\), \(g=f\) or \(g=-\text{sin }z+\kappa\pi\), \(f=\text{sin }z+\lambda\pi\); (ii) when \(\deg Q=1\), \(g=f+2\kappa\pi\) or \(-f+2\kappa\pi\) for some integer \(\kappa\); (iii) when \(\deg Q>1\) and \(Q(z)\not\equiv-Q(- z)\), then \(g=f\); (iv) when \(\deg Q>1\) and \(Q(z)\equiv-Q(-z)\), then \(g=f\) or \(-f\).