Problem 1623 (difficulty: 3/10)

An entire function \(\displaystyle f(z)\) satisfies \(\displaystyle |f(1/n)|=1/n^2\) for \(\displaystyle n=1,2,\ldots\), and \(\displaystyle |f(i)|=2\). What are the possible values of \(\displaystyle |f(-i)|\)?

Solution:

Let \(\displaystyle g(z)=f(z)\cdot\overline{f(\bar{z})}\), which also is an entire function. At the points of the form \(\displaystyle 1/n\) we have \(\displaystyle g(1/n)=f(1/n)\cdot\overline{f(1/n)}=|f(1/n)|^2=(1/n)^4\). Hence, by the Unicity Theorem, \(\displaystyle g(z)=z^4\). Then \(\displaystyle 1=|i^4|=|g(i)|=|f(i)|\cdot|f(-i)|=2|f(-i)|\), so \(\displaystyle |f(-i)|=\frac12\).

Remark. The property \(\displaystyle |g(1/n)|=1/n^2\) is satisfied by the functions of the form \(\displaystyle f(z)=z^2e^{i\varphi(z)}\) where \(\displaystyle \varphi\) is an entire function whose values are real along the real axis.

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