Levin's derivative-zero problem

Definitions and mathematical statement

Let $\mathbb H_-=\{z\in\mathbb C:\operatorname{Im}z\leq0\}$. Suppose $f$ is a nonzero entire function such that every zero of every nonidentically-zero derivative $f^{(n)}$, $n=0,1,2,\ldots$, belongs to $\mathbb H_-$. Identically zero derivatives are disregarded, so that polynomials are included in the natural way.

Let $\mathcal L_-$ be the compact-open closure of the polynomials whose zeros all lie in $\mathbb H_-$: membership means being a locally uniform limit on $\mathbb C$ of such polynomials. Classify all entire functions satisfying the derivative-zero condition. In particular, must every one belong either to $\mathcal L_-$ or to one of the exceptional families \[ f(z)=c e^{az},\qquad f(z)=c(e^{ibz}-a), \quad c,a\in\mathbb C,\quad b\in\mathbb R? \] Only members of these families satisfying the original zero-location hypothesis are relevant.

This uses the broadened exceptional family in the author's September 2026 revision: the subtracted constant is allowed to be complex, rather than restricted to the unit circle. More generally, the classification target is to describe precisely any exceptional family outside $\mathcal L_-$ if the displayed list is incomplete. The half-plane is fixed for all derivative orders. Separate half-planes chosen for different derivatives, or conditions on only finitely many derivatives, would give weaker problems.

Short English statement

Classify entire functions whose zeros and all derivative zeros stay in one closed half-plane.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700017 (AMR-036-0017), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Eremenko - My favorite unsolved problems, Problem of B. Ya. Levin. Original source indexed by AMR-036-0017. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, A problem of B. Ya. Levin. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/levin.pdf.