Research Problems in Function Theory — Problem 2.75

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 $f$, put $M(r,f)=\max_{|z|=r}|f(z)|$, $T(r,f)=(2\pi)^{-1}\int_0^{2\pi}\log^+|f(re^{i\theta})|\,d\theta$, and $\rho(f)=\limsup_{r\to\infty}\log\log M(r,f)/\log r$, where $\log^+t=\max(0,\log t)$. Let $\rho(r)$ be a finite proximate order: it is eventually nonnegative, continuous and piecewise continuously differentiable, $\rho(r)\to\rho\in[0,\infty)$, and $r\rho'(r)\log r\to0$ wherever the derivative is defined. Say that $f$ has proximate order $\rho(r)$ when $0<\limsup_{r\to\infty}\log M(r,f)/r^{\rho(r)}<\infty$. Define its indicator by \[h(\theta)=\limsup_{r\to\infty}\frac{\log|f(re^{i\theta})|}{r^{\rho(r)}}.\]

Statement recorded in the supplied source.

Determine all indicator functions $h$ that can arise from an entire, locally uniformly convergent exponential series \[f(z)=\sum_{n=1}^{\infty}a_n e^{\lambda_n z},\qquad a_n>0,\quad0\leq\lambda_1<\lambda_2<\cdots\to\infty,\] of finite proximate order. [S2]

Short English statement

Classify the possible growth indicators of entire exponential series with positive coefficients and strictly increasing nonnegative exponents.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302075 (AMR-022-2075). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, arXiv:1809.07200v2 (2018), Problem 2.75. https://arxiv.org/abs/1809.07200.