Spring 2015 - Problem 8

Solution.

Let $M = \sup_{x \in \R} \set{\abs{f\p{x}}^2 + \abs{f\p{ix}}^2}$. Recall that a Phragmén-Lindelöf function for subharmonic functions on a quarter-plane is $\abs{z}^k$ for $0 < k < 2$. In particular, $\abs{z}$ is a PL function.

Observe that by assumption, $\log{\abs{f\p{z}}} \leq \abs{z}$ and so $\log{\abs{f\p{z}}}$ obeys the maximum principle on any quarter-plane. Thus, on each quadrant, we have $\log{\abs{f\p{z}}} \leq \log{M}$, since the boundary of each quadrant is contained in the coordinate axes. Hence, we see that $\log{\abs{f\p{z}}} \leq \log{M}$ in the entire plane, and so $f$ itself is bounded on $\C$. By Liouville's theorem, $f$ is constant, as desired.