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

Abstract

Let $f(z)={\sum }_{n\ge 0}{a}_n{z}^n$ be a transcendental entire function and write $M(r,f):={\max }_{|z|=r}|f(z)|$ and $\hat{I}¼(r,f):={\max }_{n\ge 0}|a_n|r^n$. A problem of Erdős asks for the value of $$B:=\underset{f}{\sup }\underset{r\to \infty }{\mathrm{liminf}}\frac{\hat{I}¼(r,f)}{M(r,f)}.$$ In 1964, Clunie and Hayman proved that $\frac{4}{7}<B<\frac{2}{\ddot{I}}€$. 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 $\frac{4}{7}$.

12 pages. Comments and suggestions are welcome