A Liouville-type result for lacunary power series and converse results for universal holomorphic functions

Abstract

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

Integral transforms and probability distributions involving a generalized hypergeometric function

“…Since ϕ ∈ U (D), there exists a strictly monotonic increasing sequence (n k ) k∈N 0 in N 0 so that s n k (ϕ) → 0 compactly on z 0 • G as k → ∞. By Theorem 1.1 and Remark 1.2 of[MM06]…”

“…Let for every α ∈ (0, π) there exist a ξ ∈ C \ {0} withξ • G e −π , e π ; α ∩ D = ∅. (6.1) If ϕ ∈ U (D), then ∆(Λ(ϕ)) = 1.Proof: The proof is a slight modification of the proof of Theorem 2.4 in[MM06]. Assume that d := ∆(Λ(ϕ)) < 1.…”