Spring 2021 - Problem 12

Jensen's formula

Find all entire functions $\func{f}{\C}{\C}$ that satisfy the following two properties:

Hint: $f\p{z} = z^3$ is one of them.

Let $f$ be such a function, and let $m \geq 0$ be such that

g\p{z} = \frac{f\p{z} - z^3}{z^m}

is still entire. Notice that $g$ is still entire of second order and that

g\p{n^{1/3}} = \frac{f\p{n^{1/3}} - n}{n^{m/3}} = 0

for all $n \in \N$ . Suppose that $g$ is not identically zero, so that we may apply Jensen's formula:

\log\,\abs{g\p{0}} + \sum_{\abs{a_n} < R} \log\frac{R}{\abs{a_n}} = \frac{1}{2\pi} \int_0^{2\pi} \log\,\abs{g\p{Re^{i\theta}}} \,\diff\theta \lesssim R^2

where $a_n$ denotes the zeroes of $g$ . Then because each $n^{1/3}$ is a root,

\begin{aligned} R^2 &\gtrsim \sum_{n^{1/3} < R} \log\frac{R}{n^{1/3}} \\ &\gtrsim \sum_{n < R^3} \log\frac{R}{R^{1/3}} \\ &\gtrsim \floor{R^3} \log{R}, \end{aligned}

which is impossible. Hence, $g$ must have been identically zero to begin with, i.e., $f\p{z} = z^3$ , so $z^3$ is the only such function.