**Problem:** *Let be entire and suppose that for all : *

*Show that .*

**Solution:** For let

On one hand, using the fact that

we get that as , for each .

And on the other hand, direct integration shows that where . .

**Questions:** Is there a different proof, say, using the maximum principle? What if is simply harmonic?

**Update:** The same power series approach seems to give the harmonic case.

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Here’s another proof (for entire functions): u=log|f| is subharmonic, and its averages over circles |z-z_0|=R tend to -infinity as R->infinity. (Because the singularity at Re z=0 is now negligible). Not as slick, of course.

Ah yes! I was looking at the harmonic case and didn’t see this simple argument.

As an aside, I didn’t know before that is actually subharmonic. In fact it seems that , which is positive unless .

Oops I guess there are some integrability issues with ….