Research Problems in Function Theory — Problem 2.19

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.

Fix $A,B>0$ and $K\ge0$. Let $\mathcal E(A,B,K)$ be the class of entire functions for which there is a constant $C_f>0$ satisfying \[ |f(z)|\le C_f e^{K|z|}\quad(z\in\mathbb C),\qquad |f(x)|\le A\quad(x<0),\qquad |f(x)|\le B\quad(x>0). \] Thus $K$ bounds the exponential type, while the multiplicative constant in the global growth bound may depend on $f$.

Statement recorded in the supplied source.

Find the sharp pointwise bound \[\Phi_{A,B,K}(z)=\sup\{|f(z)|:f\in\mathcal E(A,B,K)\}\] in terms of $A,B,K$ and $z$. [S2] For equal half-line bounds the familiar bound is $|f(x+iy)|\le A e^{K|y|}$. The problem asks for the unequal-bound case; Update 2.19 distinguishes a subharmonic upper estimate from the sharp entire-function problem. [S2]

Short English statement

Determine the best possible complex-plane bound for an entire function of bounded exponential type when its bounds on the two real half-lines may differ.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302019 (AMR-022-2019). 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 2.19 and Update 2.19. https://arxiv.org/abs/1809.07200.