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 \(f,f',f'',\dots\) are dense in the space \(H(\mathbb{C})\) of entire functions with the topology of locally uniform convergence; \(\mathrm{HC}(D)\) denotes the set of these functions. G. Costakis proved (Oberwolfach Report 6/2008, Theorem 6) that there is a residual set \(G\subset\mathrm{HC}(D)\) such that \(pf\in\mathrm{HC}(D)\) for every \(f\in G\) and every non-zero polynomial \(p\), and asked (Question 5 there) whether \(pf\in\mathrm{HC}(D)\) for every \(f\in\mathrm{HC}(D)\) and every non-constant polynomial \(p\). We show that the answer is no. Theorem A gives an explicit \(f_A=\sum_m a_mz^m/m!\) in \(\mathrm{HC}(D)\), built from the Taylor data of an enumeration of the polynomials with non-zero Gaussian-rational coefficients placed in sparse blocks at positions \(n_j\ge2/\delta_j\), where \(\delta_j\) bounds the moduli of the data in block \(j\) from below. Then \((zf_A)^{(n)}(0)=na_{n-1}\) is \(0\) or of modulus at least \(2\) for every \(n\), so no derivative of \(zf_A\) lies in the open set \(\{h:|h(0)-1|<1\}\), and \(zf_A\notin\mathrm{HC}(D)\); the same \(f_A\) works for every monomial \(cz^d\) with \(c\ne0\), \(d\ge1\). Theorem B gives, with a ramp of large Taylor coefficients in front of each block, one \(f_B\in\mathrm{HC}(D)\) with \(pf_B\notin\mathrm{HC}(D)\) for every non-constant polynomial \(p\): beyond a threshold depending on \(p\), \((pf_B)^{(n)}(0)\) 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 \(\mathrm{HC}(D)\). 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 \((pf)^{(n)}(0)=\sum_ic_i\,n(n-1)\cdots(n-i+1)\,a_{n-i}\) for \(p=\sum_ic_iz^i\) 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.