Research Problems in Function Theory — Problem 1.10

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. The order and lower order in the plane are respectively $\rho=\limsup_{r\to\infty}\log T(r,f)/\log r$ and $\lambda=\liminf_{r\to\infty}\log T(r,f)/\log r$. Assume $\rho<\infty$ and that there are three distinct values $a_1,a_2,a_3\in\widehat{\mathbb C}$ with $\delta(a_j,f)>0$. The values may include $\infty$; their deficiencies need not have a common prescribed lower bound. Is it true that for every fixed dilation factor $\sigma>1$, \[ \limsup_{r\to\infty}\frac{T(\sigma r,f)}{T(r,f)}<\infty? \] Equivalently, for each $\sigma$ there should be constants $C_{f,\sigma}<\infty$ and $r_{f,\sigma}$ such that $T(\sigma r,f)\le C_{f,\sigma}T(r,f)$ whenever $r\ge r_{f,\sigma}$. The conclusion is required on all sufficiently large radii. An estimate after removing an exceptional set of radii is weaker than the requested assertion.

Short English statement

Do three deficient values prevent a finite-order meromorphic function's characteristic from making arbitrarily large jumps under a fixed radial dilation?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2301010 (AMR-022-1010), 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 1.10. Original source indexed by AMR-022-1010. https://arxiv.org/abs/1809.07200.