Not a complete solution, only a remark: assume $G(z)$ has no zeroes in the closed unit disk. If the series for $G(z)$ has radius of convergence >1 the series for $\log(G(z)$ will converge absolutely in $z=1$. (Because $G^\prime(z)/G(z)$ is then regular in a disk slightly larger than the unit disk.) Hence counterexamples (if any) must have a singularity in $z=1$.