Research Problems in Function Theory — Problem 2.39

Definitions and mathematical statement

For a transcendental entire function $f$, let \[ T(r,f)=\frac1{2\pi}\int_0^{2\pi}\log^+|f(re^{i\theta})|\,d\theta, \qquad m_0(r,f)=\min_{|z|=r}|f(z)|, \] where $\log^+x=\max(0,\log x)$. Its lower order is $\lambda(f)=\liminf_{r\to\infty}\log T(r,f)/\log r$. For each prescribed finite $\lambda\ge1$, determine the sharp constant \[ D(\lambda)=\inf_{\lambda(f)=\lambda} \limsup_{r\to\infty}\frac{\log m_0(r,f)}{T(r,f)}. \] This is the largest number for which every entire function of that lower order satisfies the corresponding lower bound for the upper limit. Zeros on a circle contribute $\log m_0=-\infty$, and do not remove that circle from the definition. The smaller-order range is supplied as context by the source: $D(\lambda)=\pi\lambda\cot(\pi\lambda)$ for $0\le\lambda<1/2$, with the limiting value at zero, and $D(\lambda)=\pi\lambda\cos(\pi\lambda)$ for $1/2\le\lambda<1$. The task is the unresolved sharp lower-order range at and above one; a lower bound without matching extremal examples does not determine $D(\lambda)$.

Short English statement

Find the sharp lower bound for the normalized logarithmic minimum modulus of an entire function with prescribed lower order at least one.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2302039 (AMR-022-2039), 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 2.39. Original source indexed by AMR-022-2039. https://arxiv.org/abs/1809.07200.