Research Problems in Function Theory — Problem 2.17

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For an entire function $f$, write \[ M(r,f)=\max_{|z|=r}|f(z)|,\qquad m_0(r,f)=\min_{|z|=r}|f(z)|. \] For nonconstant $f$, its order is $\rho(f)=\limsup_{r\to\infty}\log\log M(r,f)/\log r$; the lower order is defined using $\liminf$ instead. Every constant entire function, including the zero function, has order and lower order zero. In logarithmic modulus ratios, use the extended-real convention $\log0=-\infty$. Exclude monomials $f(z)=az^n$ and let $f$ be a nonconstant entire function. At a radius where the derivatives exist, define \[b_f(r)=\left(r\frac{d}{dr}\right)^2\log M(r,f). \] Limits below are taken through such radii; the one-sided limits at exceptional radii have the convention used in Problem 2.18. Define the universal sharp constant \[A_* = \inf_f\limsup_{r\to\infty}b_f(r),\] with the infimum over this class of functions.

Statement recorded in the supplied source.

Determine $A_*$. In particular, is the originally proposed value $A_*=1/4$ correct? [S2] Update 2.17 explicitly excludes monomials and reports Kjellberg's disproof of $A_*=1/4$, with $0.24<A_*<0.248$. [S2]

Short English statement

Find the largest universal lower bound on the eventual curvature of the logarithm of an entire function's maximum modulus, after excluding monomials.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302017 (AMR-022-2017). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter 2, Problem 2.17 and Update 2.17. https://arxiv.org/abs/1809.07200.