Research Problems in Function Theory — Problem 1.41

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 $(a_j)$ be a finite or countable sequence of distinct points of $\widehat{\mathbb C}$, and let prescribed numbers $d_j\in(0,1]$ satisfy $\sum_jd_j\le2$. The disc inverse-deficiency problem asks for a meromorphic function $f$ on $\mathbb D$ such that \[ \limsup_{r\uparrow1}\frac{T(r,f)}{\log(1/(1-r))}=\infty, \quad \delta(a_j,f)=d_j\quad\text{for every }j, \] and $\delta(a,f)=0$ at every other value $a\in\widehat{\mathbb C}$. The infinity value is treated as the pole value, not as a limit of the finite prescribed points. One may equivalently prescribe only finite values and prescribe the deficiency at infinity separately. Are the individual restrictions $d_j\le1$ and the total bound two sufficient for all such data? Exact values and absence of extra deficient values are part of the question; constructing lower bounds $\delta(a_j,f)\ge d_j$ alone does not answer it.

Short English statement

Can every admissible list of deficiencies be realized exactly by a sufficiently rapidly growing meromorphic function in the unit disc?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2301041 (AMR-022-1041), 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 1.41. Original source indexed by AMR-022-1041. https://arxiv.org/abs/1809.07200.