Polynomial-Exponential Counterexamples to a Derivative-Zero Half-Plane Classification

Let \(P\) be the compact-open closure of the polynomials whose zeros lie in the closed lower half-plane. The entire function \(e^{iz}+z+2i\) has every zero of every derivative in that half-plane, but lies neither in \(P\) nor in the exceptional families \(Ce^{az}\) and \(C(e^{ibz}-a)\), with complex \(C,a\) and real \(b\). This gives a negative answer to the precise alternative in Eremenko's version of a problem of B. Ya. Levin and in the frozen UnsolvedMath record AMR-036-0017.

For every positive integer \(d\), an explicit affine holomorphic \(d\)-parameter family gives the stronger property that every derivative is zero-free in the closed upper half-plane, while remaining outside \(P\) and the stated exceptional families. These exceptions cannot be covered by countably many holomorphic families of uniformly bounded finite dimension. The proof uses polynomial dominance and a logarithmic-derivative condition. Buhovski's earlier exponential example is credited.

This is not a classification of all admissible entire functions. The preprint is AI-assisted, self-audited and unrefereed, with no independent review, formal verification or absolute priority claim.