Generalizing the Clunie--Hayman construction in an Erdős maximum-term problem

TL;DR

The paper tackles the Erd\H{o}s maximum-term problem for transcendental entire functions by generalizing the Clunie–Hayman construction with a two-parameter family $f_{K,\varepsilon}$ and its bilateral companion $k_{K,\varepsilon}$. A central scaling identity $k_{K,\varepsilon}(Kz)=z\,k_{K,\varepsilon}(\varepsilon z)$ yields an exact relation $\beta(f_{K,\varepsilon})=1/A(K,\varepsilon)$, where $A(K,\varepsilon)=\max_{|z|=1}|k_{K,\varepsilon}(z)|$, reducing the problem to minimizing a unit-circle maximum. The authors establish an explicit numerical lower bound by selecting $K_0$ and $\varepsilon_0$ and certifying that $A_0=\max_{|z|=1}|k_{K_0,\varepsilon_0}(z)|<1.70919$, which implies $\beta(f_{K_0,\varepsilon_0})>0.58507$ and hence $B>0.58507$. The result improves the classical lower bound $4/7\approx0.57143$ and demonstrates how theta-function connections and certified numerics can sharpen bounds in this extremal problem with implications for the distribution of power-series terms relative to the maximum modulus.

Abstract

Let $f(z)=\sum_{n\ge0}a_n z^n$ be a transcendental entire function and write $M(r,f):=\max_{|z|=r}|f(z)|$ and $μ(r,f):=\max_{n\ge0}|a_n|\,r^n$. A problem of Erdős asks for the value of $$ B:=\sup_f \liminf_{r\to\infty}\frac{μ(r,f)}{M(r,f)}. $$ In 1964, Clunie and Hayman proved that $\frac{4}{7}<B<\frac{2}π$. In this paper we develop a generalization of their construction via a scaling identity and obtain the explicit lower bound $$ B>0.58507, $$ improving the classical constant $\frac{4}{7}$.

Theorems & Definitions (32)

• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Proposition 2.5