Polynomial Multiples of MacLane's Universal Functions Need Not Be Universal: A Negative Answer to a Question of Costakis

An entire function f is universal in the sense of MacLane if its derivatives f, f′, f″, … are dense in the space H(ℂ) of entire functions with the topology of locally uniform convergence; HC(D) denotes the set of these functions. G. Costakis proved (Oberwolfach Report 6/2008, Theorem 6) that there is a residual set G ⊂ HC(D) such that pf ∈ HC(D) for every f ∈ G and every non-zero polynomial p, and asked (Question 5 there) whether pf ∈ HC(D) for every f ∈ HC(D) and every non-constant polynomial p. We show that the answer is no. Theorem A gives an explicit f_A = Σ_m a_m z^m/m! in HC(D), built from the Taylor data of an enumeration of the polynomials with non-zero Gaussian-rational coefficients placed in sparse blocks at positions n_j ≥ 2/δ_j, where δ_j bounds the moduli of the data in block j from below. Then (z f_A)^(n)(0) = n an−1 is 0 or of modulus at least 2 for every n, so no derivative of z f_A lies in the open set {h : |h(0) − 1| < 1}, and z f_A ∉ HC(D); the same f_A works for every monomial c z^d with c ≠ 0, d ≥ 1. Theorem B gives, with a ramp of large Taylor coefficients in front of each block, one f_B ∈ HC(D) with p f_B ∉ HC(D) for every non-constant polynomial p: beyond a threshold depending on p, (p f_B)^(n)(0) is 0 or of modulus at least 2, and finitely many derivatives do not affect density. Hence Costakis's Theorem 6 cannot be extended from a residual set to all of HC(D). The question and the residual result are Costakis's, and the construction is the classical block construction of MacLane-type universal functions; the only new observation is that the right-hand side of the Leibniz formula (pf)^(n)(0) = Σ_i c_i n(n−1)⋯(n−i+1) an−i for p = Σ_i c_i z^i can be controlled on such functions (with the ramps for Theorem B). Gardiner (2014) answered the analogous questions for universal Taylor series negatively. The argument is elementary; we found no printed answer to the question, but a folklore version may exist. Exact computations illustrate the constructions. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-1787-006 (G. Costakis, Which maps preserve universal functions?, Oberwolfach Report 6/2008, pp. 328–331, Question 5).