Research Problems in Function Theory — Problem 2.33

Definitions and mathematical statement

For a transcendental entire function $f$, write $M(r)=\max_{|z|=r}|f(z)|$ and $m(r)=\min_{|z|=r}|f(z)|$; set $\log m(r)=-\infty$ if the circle meets a zero. Its lower order is $\lambda(f)=\liminf_{r\to\infty}\log\log M(r)/\log r$, and its order uses $\limsup$ instead. For finite $\lambda$, define the optimal universal constant \[ C(\lambda)=\inf_{\lambda(f)=\lambda} \limsup_{r\to\infty}\frac{\log m(r)}{\log M(r)}. \] For functions of infinite order define analogously \[ C_\infty=\inf_{\rho(f)=\infty} \limsup_{r\to\infty} \frac{\log m(r)}{\log M(r)\,\log\log\log M(r)}. \] All logarithms in the latter denominator are defined and the denominator is positive at sufficiently large radii. The problem asks for the exact value of $C_\infty$ and the sharp asymptotic behaviour of $C(\lambda)/\log\lambda$ as $\lambda\to\infty$, including whether this ratio has a limit. The infima express universal lower bounds, not bounds for a particular function. The source supplies nonmatching negative bounds and leaves the optimal constants undetermined.

Short English statement

Find the sharp large-growth constants controlling how small an entire function can remain on complete circles relative to its maximum modulus.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2302033 (AMR-022-2033), 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 2.33. Original source indexed by AMR-022-2033. https://arxiv.org/abs/1809.07200.