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Spring 2013 - Problem 7

subharmonic functions

Let $\func{f}{\C}{\C}$ be an entire function such that $\log\,\abs{f}$ is absolutely integrable with respect to planar Lebesgue measure. Show that $f$ is constant.

Solution.

Suppose $f$ is non-constant. By Liouville's theorem, $f$ cannot be bounded so in particular, there exists $z_0 \in \C$ such that $\abs{f\p{z_0}} \geq e \implies \log\,\abs{f\p{z_0}} \geq 1$ .

Notice that $\log\,\abs{f}$ is subharmonic, so by the sub-mean value property,

\begin{aligned} \int_{\C} \log\,\abs{f\p{z}} \,\diff{A} &= \int_0^\infty \int_0^{2\pi} r\log\,\abs{f\p{z_0 + re^{i\theta}}} \,\diff\theta \,\diff{r} \\ &\geq \int_0^\infty 2\pi r \,\diff{r} \\ &= \infty, \end{aligned}

so $\log\,\abs{f\p{z}}$ is not absolutely integrable.