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arXiv · 2602.12217

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

Yixin He·Quanyu Tang
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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} 0.58507, $$ improving the classical constant $\frac{4}{7}$.

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Yixin He, Quanyu Tang. 2026-02-12. Generalizing the Clunie--Hayman construction in an Erd\H{o}s maximum-term problem. https://arxiv.org/abs/2602.12217

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