Approximation by strongly annular solutions of functional equations

Abstract

A major result of this paper is that the set of all functions g ( z ) g(z) such that g g is strongly annular and is a solution of a Mahler type of functional equation given by g ( z ) = q ( z ) g ( z p ) g(z)=q(z)g(z^p) where p ≥ 2 p\ge 2 is an integer and q q is a polynomial with q ( 0 ) = 1 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 ± 1 \pm 1 with the same topology.