[Paper Review] An extremal problem for a class of entire functions of exponential type

Alexandre Erëmenko, Peter Yuditskii|arXiv (Cornell University)|Jul 13, 2008

Meromorphic and Entire Functions8 references3 citations

TL;DR

This paper solves an extremal problem for entire functions of exponential type whose indicator diagram is contained in $[-i\sigma, i\sigma]$, proving that the upper density of zeros is at most $c\sigma$, where $c \approx 1.508879$ is the unique positive solution to $\log(\sqrt{c^2+1}+c) = \sqrt{1 + c^{-2}}$. The bound is sharp, with equality achieved by a specific extremal function constructed via conformal mapping and subharmonic function theory.

ABSTRACT

We find the exact upper estimate for the upper density of zeros of entire functions of exponential type whose indicator diagram is contained in a given interval.

Motivation & Objective

• To determine the best possible upper bound on the upper density of zeros for entire functions of exponential type whose indicator diagram is contained in $[-i\sigma, i\sigma]$.
• To establish that the classical bound $2e\sigma/\pi \approx 1.7305\sigma$ is not sharp within this class.
• To construct an extremal function achieving the sharp bound $c\sigma$, where $c$ solves a specific transcendental equation.
• To prove that the upper density limit exists and equals $c\sigma$ for functions in this class, using subharmonic function theory and conformal mapping.

Proposed method

• Reduction to the case $\sigma = 1$ and consideration of even functions to preserve symmetry and scale the density by a factor of 2.
• Application of the compactness principle for subharmonic functions to extract a limit function $v$ from a sequence $v_n(z) = t_n^{-1}\log|f(t_n z)|$, which inherits the growth bound $v(z) \leq |\mathrm{Im}\, z|$.
• Use of the Riesz measure $\mu$ of the subharmonic limit $v$ to relate the zero density to the mass of $\mu$ on the unit disk.
• Reduction of the problem to the case where the Riesz measure is supported on the real line via radial projection, leading to a harmonic function $v^*$ satisfying $v^*(z) \leq \sigma' |\mathrm{Im}\, z|$ with $\sigma' \leq 1$.
• Construction of a conformal map $\phi = u + iv$ from the upper half-plane onto a region $G = \{x+iy : y > g(x)\}$ with $g \leq 0$, $g(0) = 0$, and $\phi(0) = 0$, $\phi(iy) \sim iy$ as $y \to \infty$, to model the extremal behavior.
• Solution of the extremal problem by maximizing $\mathrm{Re}\, \phi(1)$ over such conformal maps, leading to a Schwarz–Christoffel integral representation and derivation of the critical equation for $c$.

Experimental results

Research questions

• RQ1 What is the best possible upper bound on the upper density of zeros for entire functions of exponential type with indicator diagram contained in $[-i\sigma, i\sigma]$?
• RQ2 Can the classical bound $2e\sigma/\pi \approx 1.7305\sigma$ be improved within this class of functions?
• RQ3 Is there a function in $E_\sigma$ for which the upper density of zeros achieves the bound $c\sigma$, and if so, what is the value of $c$?
• RQ4 What geometric and analytic conditions characterize the extremal function that achieves the sharp bound?

Key findings

• The upper density of zeros of any function $f \in E_\sigma$ is at most $c\sigma$, where $c \approx 1.508879$ is the unique positive solution to $\log(\sqrt{c^2+1} + c) = \sqrt{1 + c^{-2}}$.
• This bound is sharp: for every $\sigma > 0$, there exists a function $f \in E_\sigma$ such that the upper density of its zeros is exactly $c\sigma$.
• The extremal function is constructed via a conformal map from the upper half-plane onto a region with a vertical slit at $|x| < \pi c/2$, which corresponds to a specific jump discontinuity in the boundary function $g(x)$.
• The value $c \approx 1.508879$ is derived from the condition that the imaginary part of a Schwarz–Christoffel integral vanishes at a critical point, ensuring the extremal behavior.
• The proof relies on the subharmonic function limit $v$ with Riesz measure $\mu$, and the reduction to a harmonic function in the upper half-plane via radial projection.
• The maximal value of $\mathrm{Re}\, \phi(1)$ for the conformal map $\phi$ is achieved when the boundary function $g(x)$ is $-\infty$ on $|x| < \pi c/2$ and $0$ elsewhere, leading to the optimal zero density.