Research Problems in Function Theory — Problem 2.69

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 $\log^+t=\max(0,\log t)$. A transcendental entire function is an entire function that is not a polynomial.

Statement recorded in the supplied source.

Is there an absolute constant $d>0$ such that every transcendental entire function $f$ with $\rho(f)<d$ satisfies \[\liminf_{r\to\infty}\frac{T(r,f)}{T(r,f')}\leq1?\] [S2]

Short English statement

Does a universal positive upper bound on the order force the characteristic of an entire function to be asymptotically no larger than that of its derivative along some radii?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302069 (AMR-022-2069). 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 2.69. https://arxiv.org/abs/1809.07200.