Generalizing the Clunie--Hayman construction in an Erd\H{o}s maximum-term problem

TL;DR

This paper generalizes a classical construction in complex analysis to improve the lower bound on a supremum related to the growth of transcendental entire functions, advancing understanding of Erdős's maximum-term problem.

Contribution

It introduces a generalized construction via a scaling identity that improves the lower bound for Erdős's problem from 4/7 to approximately 0.58507.

Findings

Established a new explicit lower bound B>0.58507

Extended the Clunie--Hayman construction using a scaling identity

Improved the classical constant in Erdős's maximum-term problem

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 $\mu(r,f):=\max_{n\ge0}|a_n|\,r^n$. A problem of Erd\H{o}s asks for the value of $$ B:=\sup_f \liminf_{r\to\infty}\frac{\mu(r,f)}{M(r,f)}. $$ In 1964, Clunie and Hayman proved that $\frac{4}{7}<B<\frac{2}{\pi}$. 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 $\frac{4}{7}$.