Generalizing the Clunie--Hayman construction in an Erdős 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 $μ(r,f):=\max_{n\ge0}|a_n|\,r^n$. 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}0.58507, $$ improving the classical constant $\frac{4}{7}$.
Fields of Study & Subjects
Meromorphic and Entire Functions
Holomorphic and Operator Theory
Mathematical Dynamics and Fractals
Publication details
- Journal / source
- arXiv (Cornell University)
- Publication date
- 12 February 2026
- Language
- EN
- Document type
- Preprint