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Fall 2018 - Problem 7

harmonic functions

Let $f\p{z}$ be an analytic function on the entire complex plane $\C$ such that the function $U\p{z} = \log\,\abs{f\p{z}}$ is Lebesgue area integrable: $\int_\C \abs{U\p{z}} \,\diff{x} \,\diff{y} < \infty$ . Prove $f$ is constant.

Solution.

If $U$ is identically zero, then the claim is clear. Otherwise, suppose there exists $z_0 \in \C$ with $U\p{z_0} \neq 0$ . Then because $U\p{z}$ is subharmonic,

\begin{aligned} \int_\C U\p{z} \,\diff{x} \,\diff{y} &= \int_0^\infty r \int_0^{2\pi} u\p{z_0 + re^{i\theta}} \,\diff\theta \,\diff{r} \\ &\geq \int_0^\infty 2\pi r u\p{z_0} \,\diff{r} \\ &= \infty, \end{aligned}

which is impossible. Thus, $U$ must have been identically zero to begin with.