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I was reading some slides from a lecture. In a proof, there arose the need to show a certain function $f : \mathbb{C} \to \mathbb{C}$ was constant. The argument proceeded by checking that

  1. $f$ was entire
  2. $f(z+i) = f(z)$ for all $z$
  3. $f$ was bounded on $\mathbb{R}$

and then concluding that $f$ must be constant. I followed the proofs of the three claims no problem, but my complex analysis is weak enough that I'm unsure how one is supposed to conclude that $f$ is constant. Clearly Liouville's theorem does not apply directly. My guess is that some kind of boundary principle is being applied to the "rectangle" $R = \{ z \in \mathbb{C} : 0 \leq \Im(z) \leq 1 \}$. For instance, if the maximum of $|f|$ on $R$ must occur on the boundary, then the result follows. My complex analysis is patchy enough, though, that I'm unaware if such a result.

Added:

  • Relevant: Lindelof's theorem
  • The above article contains an enlightening example. If we take $f(z) = \exp(\exp(2 \pi z))$, then (1) and (2) are satisfied. (3) is "half satisfied" in the sense that $\lim_{t \to -\infty} f(z) = 1$ here.
Mike F
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  • I've just seen the edit you added while I was trying to find a counterexample...that's pretty close! Just put a minus sign and the function becomes bounded on $\mathbb{R}$. – Julien Apr 29 '13 at 02:56
  • @julien: Haha! Wow you're right. I was pretty dubious when I first read this "...pfffff what is this character talking about, how could introducing a minus sign affect anything..." but, yes, that works great! Nice work. – Mike F Apr 29 '13 at 03:01
  • And I first thought there was a theorem proving boundedness... – Julien Apr 29 '13 at 03:04

1 Answers1

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You need more assumptions to conclude that $f$ is constant. For instance, the function $$ f(z)=\exp(-\exp(2\pi z)) $$ satisfies all your assumptions.

Note: I had not thought about that in a long time, and I first thought Hadamard's three-line theorem would prove boundedness on the strip. But my memory was playing me some trick, and boundedness on the strip is actually a requirement of the latter.

Julien
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