Non-constant entire function-bounded or not?

Question

Show that if $f$ is a non-constant entire function,it cannot satisfy the condition: $$f(z)=f(z+1)=f(z+i)$$

My line of argument so far is based on Liouville's theorem that states that every bounded entire function must be a constant.

So I try-to no avail-to show that if $f$ satisfies the given condition, it must be bounded. I haven't made much progress with this, so any hints or solutions are welcome.

AlexR · Accepted Answer · 2015-05-21T08:26:17.140

4

First you can prove that $$f(z+\mathbb Z(i)) = f(z)$$ so $f$ is determined by it's values on the square $[0, 1+i)$. Entire functions are bounded on precompact sets. QED.

edited May 21 '15 at 08:26

answered May 21 '15 at 08:22

AlexR

25,110

What is meant here by "entire functions on precompact domains"? By definition, the domain of an entire function is all of $\Bbb C$. – Travis Willse May 21 '15 at 08:24
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@Travis Entire functions are holomorphic on every subset of $\mathbb C$. This allows us to conclude boundedness on any precompact subset of $\mathbb C$. – AlexR May 21 '15 at 08:26
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This wording is much clearer, cheers! – Travis Willse May 21 '15 at 09:24

Non-constant entire function-bounded or not?

1 Answers1