On the growth of universal functions

In this paper, we study the growth of universal functions (Taylor series) on the unit discD. We describe a class of growth types prohibited for such a function. By investigating the relation between growth and value distribution, we prove that every universal function assumes every complex value, with at most one exception, on sequences inD that approach ϖD rather slowly. We use this to get a large class of equations, including polynomials with Nevanlinna coefficients and equations of iterates, that no universal function satisfies. Finally, we produce a universal function whose growth is bounded by, which is close to the rates of growth prohibited. The author would like to thank the referee for suggesting references [3] and [4].