Research Problems in Function Theory — Problem 2.53

Definitions and mathematical statement

For entire functions $f,g$, write $f\preceq g$ if every sequence $(z_n)$ with $|f(z_n)|\to\infty$ also satisfies $|g(z_n)|\to\infty$. For meromorphic functions use the same condition at points where both functions are finite. This is a preorder; functions with $f\preceq g\preceq f$ are considered equivalent. The source asks three related questions. If $f$ is entire and $f''\preceq f$, what growth restrictions follow? In particular, must the spherical derivative \[ f^{\#}(z)=\frac{|f'(z)|}{1+|f(z)|^2} \] be bounded on $\mathbb C$? This boundedness is the plane notion of a normal meromorphic function, not the disc condition with a boundary-distance factor. Do any two entire functions admit a greatest lower bound and a least upper bound in this preorder? For instance a greatest lower bound $h$ must satisfy $h\preceq f,g$, and $k\preceq f,g$ must imply $k\preceq h$. The dual inequalities define a least upper bound; uniqueness is only up to equivalence. Finally, for meromorphic $f$ satisfying $f'\preceq f$, determine a universal growth restriction, with order at most two proposed as a possible conclusion. For nonconstant $f$, its order is $\limsup_{r\to\infty}\log T(r,f)/\log r$, where, for $r\ge1$, \[ T(r,f)=\frac1{2\pi}\int_0^{2\pi}\log^+|f(re^{it})|\,dt +m_0\log r+\sum_{0<|a|\le r}m_a\log\frac r{|a|}. \] Here the sum runs over the nonzero poles, $m_a$ is the pole multiplicity at $a$ (zero if absent), and $\log^+x=\max(0,\log x)$. Constants have order zero.

Short English statement

Study derivative growth and lattice operations for the ordering that compares where analytic functions tend to infinity.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2302053 (AMR-022-2053), 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 2.53. Original source indexed by AMR-022-2053. https://arxiv.org/abs/1809.07200.