Problem 227

Let $f=\sum_{n=0}^\infty a_nz^n$ be an entire function which is not a polynomial. Is it true that if

\lim_{r\to \infty} \frac{\max_n\lvert a_nr^n\rvert}{\max_{\lvert z\rvert=r}\lvert f(z)\rvert}

exists then it must be $0$ ?

Source: erdosproblems.com/227

Analysis

Wiki page

Status

SolvedDisproved

An accepted solution exists. The statement is false.

Disproved. The site credits Clunie and Hayman [ClHa64], who produce transcendental entire functions for which the ratio tends to any prescribed value in $[0,1/2]$ ; the claim page Clunie and Hayman 1964 records it, accepted on its refereed publication and the site's credit.

Problem 227

Status

Proof claims

Discussion