Generalizing the Clunie--Hayman construction in an Erd\H{o}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 $\frac{4}{7}$.