Research Problems in Function Theory — Problem 1.13

Definitions and mathematical statement

For a nonconstant entire function $f$, let $M(r,f)=\max_{|z|=r}|f(z)|$ and define order and lower order by \[ \rho(f)=\limsup_{r\to\infty}\frac{\log\log M(r,f)}{\log r},\qquad \lambda(f)=\liminf_{r\to\infty}\frac{\log\log M(r,f)}{\log r}. \] Only sufficiently large $r$ are used. For an entire function the Nevanlinna characteristic is \[ T(r,f)=\frac1{2\pi}\int_0^{2\pi}\log^+|f(re^{i\theta})|\,{\,\mathrm{d}}\theta. \] Let $N(r,0;f)$ be the integrated zero-counting function with multiplicities: for $r>1$ it is the sum of $\log(r/|a|)$ over nonzero zeros $a$ with $|a|<r$, plus $m_0\log r$, where $m_0$ is the multiplicity of the zero at the origin. The deficiency at zero is \[ \delta(0,f)=1-\limsup_{r\to\infty}\frac{N(r,0;f)}{T(r,f)}. \]

For each realizable parameter pair $0\leq\lambda\leq\rho<\infty$, let $\mathcal E_{\rho,\lambda}$ be the class of nonconstant entire functions having order exactly $\rho$, lower order exactly $\lambda$, and all zeros real. Define the best universal lower bound by \[ B(\rho,\lambda)=\inf_{f\in\mathcal E_{\rho,\lambda}}\delta(0,f), \qquad\mathcal E_{\rho,\lambda}\ne\varnothing. \] The problem is to determine this sharp bound. The source attributes to Edrei, Fuchs and Hellerstein the positivity $B(\rho,\lambda)>0$ for $2<\rho<\infty$, and conjectures that $B\to1$ as $\rho\to\infty$.

Because $B$ also depends on $\lambda$, the latter statement needs an additional quantifier: the record does not say whether $\lambda$ is fixed, varies in a specified way with $\rho$, or whether convergence is uniform over realizable lower orders. These alternatives are retained as an unresolved asymptotic qualification.

Short English statement

Find the sharp guaranteed deficiency of zero for finite-order entire functions whose zeros are all real, with both order and lower order prescribed. The source reports positive deficiency bounds when the order exceeds two and conjectures approach to one at large order, with lower-order dependence unspecified. The supplied triage says the problem was open in Hayman's 2018 edition and no recent resolution was located; that is a dated literature-search report, not an independent current-status conclusion.

Sources

• UnsolvedMath permanent ID 2301013, source code catalogue entry, "Research Problems in Function Theory --- Problem 1.13." Registry curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0; updated 2026-08-06.
• Hayman, Research Problems in Function Theory (2018 source update), Problem 1.13: https://arxiv.org/abs/1809.07200. Supplied source-TeX extraction accessed 2026-07-29. The metadata retains the 2018 update for status triage and marks status/release and rights review NEEDS_REVIEW.
• Edrei, Fuchs and Hellerstein are named in the supplied statement for the positive-bound result, without complete titles, dates, theorem numbers or locators. No missing reference is inferred.