No abstract

We prove the existence of a function j which is holomorphic exactly in the unit disk D and has universal translates with respect to a prescribed closed set E … "D and satisfies j ¥ C Key Words: universal functions; gap power series; boundary behaviour of holomorphic functions; lacunary polynomial approximation.

In this article the existence of meromorphic functions on C is investigated which are universal under prescribed translates and whose Nevanlinna's characteristic has slow transcendental growth.

A Liouville-type result is proved for holomorphic functions having lacunary power series with restricted growth for a subsequence of the partial sums. As an application, converse results concerning existence of universal holomorphic functions are given. A Liouville-type resultLiouville's theorem states that an entire function ϕ reduces to a polynomial of degree at most [γ ] (where [γ ] is the integral part of the real number γ ) if |ϕ(z)| ≤ M|z| γ for some M > 0. The conclusion of Liouville's theorem holds more generally if ∞ ν=0 ϕ ν z ν is a (formal) power series with partial sums s n (z) such that for some subsequence {k m } of the positive integers and some M > 0 we have |s k m (z)| ≤ M|z| γ for z ∈ E m , where {E m } is an exhaustion of the complex plane. Let in the sequel Q always be a subset of the nonnegative integers. We shall prove a Liouville-type result for lacunary power series ϕ(z) = ν∈Q ϕ ν z ν , (1.1) in which we replace the sets E m by appropriate smaller subsets of the plane which depend on Q. We first recall some notions about entire functions. If Q = {q j : j = 1, 2, . . . } then f(w) := ∞ j=1 1 − w 2 q 2 j AMS 2000 subject classifications: 30B10

This paper is devoted to the converse of the classical Fabry-Pólya theorem dealing with the localization of singularities of lacunary power series on the boundary of the circle of convergence.