Research Problems in Function Theory — Problem 1.7

Definitions and mathematical statement

For a nonconstant meromorphic function $f$, let $n(t,a)$ count solutions of $f(z)=a$ in $|z|\le t$, with multiplicity; $a=\infty$ means poles. Put \[ N(r,a)=\int_0^r\frac{n(t,a)-n(0,a)}t\,dt+n(0,a)\log r, \quad T(r,f)=N(r,\infty)+\frac1{2\pi}\int_0^{2\pi}\log^+|f(re^{i\theta})|\,d\theta, \] where $\log^+x=\max(0,\log x)$. At a boundary radius $R\in\{1,\infty\}$ with $T(r,f)\to\infty$, define the deficiency $\delta(a,f)=1-\limsup_{r\to R}N(r,a)/T(r,f)$. A deficient value has positive deficiency. Let $f$ be entire of prescribed finite nonintegral order $\rho=\limsup_{r\to\infty}\log T(r,f)/\log r>1$. The sum below is over the Riemann sphere, so it includes the automatic term $\delta(\infty,f)=1$. Define the largest universal defect gap by \[ K_*(\rho)=\inf_{\operatorname{ord}f=\rho}\left(2-\sum_{a\in\widehat{\mathbb C}}\delta(a,f)\right). \] With $q=\lfloor\rho\rfloor$, determine whether the sharp answer is \[ K_*(\rho)= \begin{cases} |\sin\pi\rho|/(q+|\sin\pi\rho|),&q<\rho<q+\tfrac12,\\ |\sin\pi\rho|/(q+1),&q+\tfrac12\le\rho<q+1. \end{cases} \] Thus one must both establish the proposed bound for every such function and show that no larger bound works. The source records the smaller-order ranges separately; they are not additional conjectures here.

Short English statement

Determine the sharp amount by which the total deficiencies of an entire function of nonintegral order must fall short of two.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2301007 (AMR-022-1007), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 1.7. Original source indexed by AMR-022-1007. https://arxiv.org/abs/1809.07200.