Research Problems in Function Theory — Problem 1.40

Definitions and mathematical statement

Let $\mathbb D=\{z\in{\mathbb{C}}:|z|<1\}$ and let $f$ be meromorphic in $\mathbb D$ with finite order. With the standard disk Nevanlinna notation, $T(r,f)$ is the characteristic, $N(r,a;f)$ is the integrated counting function of $a$-points with multiplicity, and $\overline N(r,a;f)$ counts distinct $a$-points. The usual disk order is \[ \rho=\limsup_{r\uparrow1}\frac{\log^+T(r,f)}{\log(1/(1-r))}. \] For functions with the required nondegenerate characteristic growth, define \[ \delta(a,g)=1-\limsup_{r\uparrow1}\frac{N(r,a;g)}{T(r,g)},\qquad k(f)=\limsup_{r\uparrow1}\frac{\overline N(r,\infty;f)}{T(r,f)+1}. \] The sum over finite deficient values means the supremum of finite partial sums over $a\in{\mathbb{C}}$.

The source attributes to Shea and Sons the inequality \[ \sum_{a\in{\mathbb{C}}}\delta(a,f) \leq \delta(0,f')\left(1+k(f)\right)+\frac{2}{\lambda}(\rho+1). \] It prints \[ \lambda(f)=\liminf_{r\to\infty} \frac{T(r,f)}{\log(1/(1-r))}. \] That limit is outside the radial domain $0<r<1$ and does not define the stated disk quantity. A plausible intended correction is $r\uparrow1$; it is explicitly conditional here, not silently made. This normalized-characteristic quantity must also not be confused with the lower order $\liminf\log^+T/\log(1/(1-r))$.

After recovering the correct definition and admissible range of $\lambda$, L. R. Sons asks whether the coefficient $2$ can be replaced by $1$, namely whether \[ \sum_{a\in{\mathbb{C}}}\delta(a,f) \leq \delta(0,f')\left(1+k(f)\right)+\frac{1}{\lambda}(\rho+1). \] The source says this improvement would be best possible. It does not state the needed $\lambda>0$ assumptions or conventions for $\lambda=0,\infty$, or the growth conditions ensuring the deficiencies of $f$ and $f'$ are defined. These must be resolved before the displayed conditional target becomes a complete theorem.

Short English statement

Sharpen the reported disk deficiency inequality by replacing the coefficient two by one in the order-dependent term. The source says that improvement would be sharp. The printed definition of $\lambda$ has an impossible radial limit for a disk function, so the intended boundary limit and parameter assumptions require review. The supplied triage says the problem was open in Hayman's 2018 update and no recent resolution was located; this draft makes no independent current-status determination.

Sources

• UnsolvedMath permanent ID 2301040, source code catalogue entry, "Research Problems in Function Theory --- Problem 1.40." 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 1.40, attributed to L. R. Sons: 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.
• Shea and Sons are named in the supplied statement for the coefficient-two inequality, without a complete bibliographic reference. No title, date, theorem number or locator is inferred.