Generalizing the Clunie--Hayman construction in an Erdős maximum-term problem
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915795036012544 |
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| author | He, Yixin Tang, Quanyu |
| author_facet | He, Yixin Tang, Quanyu |
| contents | 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 $\frac{4}{7}<B<\frac{2}π$. 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 $\frac{4}{7}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_12217 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Generalizing the Clunie--Hayman construction in an Erdős maximum-term problem He, Yixin Tang, Quanyu Complex Variables 30D15, 30D20, 65G30 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 $\frac{4}{7}<B<\frac{2}π$. 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 $\frac{4}{7}$. |
| title | Generalizing the Clunie--Hayman construction in an Erdős maximum-term problem |
| topic | Complex Variables 30D15, 30D20, 65G30 |
| url | https://arxiv.org/abs/2602.12217 |