Approximation by strongly annular solutions of functional equations

A major result of this paper is that the set of all functions $g(z)$ such that $g$ is strongly annular and is a solution of a Mahler type of functional equation given by $g(z)=q(z)g(z^p)$ where $p\ge 2$ is an integer and $q$ is a polynomial with $q(0)=1$ is a dense first category set in the set of all holomorphic functions on the open unit disk with the topology of almost uniform convergence. A second result is that strongly annular solutions of these types of functional equations are dense in the space of holomorphic functions with Maclaurin coefficients of $\pm 1$ with the same topology.