On Bounded Universal Functions

Compositionally Universal Entire Functions on the Plane and the Punctured Plane

“…We may even prescribe the set Γ within a large family, including again all lines, in the case of H(C)-universality, see Theorem 3.5. This improves recent results by A. Vogt [27] and completes a number of findings by several authors, see Section 1.…”

“…In 2010, Calderón, Luh and the author [6] stated the following: if A ⊂ C is an unbounded Arakelian set with ρ i (C \ A) = +∞, there is a dense linear manifold M of entire functions all of whose nonzero members are Birkhoffuniversal and exp(|z| α )f (z) → 0 (z → ∞, z ∈ A) for all α < 1/2 and f ∈ M . Passing to C * , A. Vogt [27] has recently constructed a multiplicative universal entire function ϕ with respect to a given unbounded sequence (a n ) such that ϕ is bounded on some curve Γ tending to ∞. On the contrary, he has proved that any such ϕ is necessarily unbounded on every line.…”

“…Without loss of generality, we may assume that (n k ) is the full sequence N. In particular, there is n 0 ∈ N such that Proof. We will follow the approach of [27] in the first steps. Choose an exhaustive sequence {K n } n≥1 in M(C * ) as in the proof of Theorem 2.1 (see [7,Lemma 2.9] or [23,Lemma 3]).…”