Research Problems in Function Theory — Problem 2.18

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 $\log0=-\infty$. For a function holomorphic on a neighbourhood of a circle $|z|=r_0>0$, define $M(r,f)$ for nearby radii and, where defined, $b_f(r)=(r\,d/dr)^2\log M(r,f)$. Write $b_f(r_0-0)$ and $b_f(r_0+0)$ for the left and right limiting values. Exclude monomials as required by Update 2.17, which Update 2.18 also invokes.

Statement recorded in the supplied source.

Can a nonmonomial entire function satisfy \[b_f(r_0-0)=b_f(r_0+0)=0\] at some $r_0>0$? More generally, can this happen for a function holomorphic on a neighbourhood of $|z|=r_0$? [S2] Update 2.18 gives a positive answer due to London, using an entire function $(1-z)\exp(\alpha z+\beta z^2+\gamma z^3)$. It distinguishes this from vanishing throughout an interval of radii. [S2]

Short English statement

Can both one-sided curvatures of the logarithmic maximum modulus vanish at the same circle, without the function being a monomial?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302018 (AMR-022-2018). 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 2.18 and Update 2.18. https://arxiv.org/abs/1809.07200.