A question around Liouville's Theorem

Asked Oct 28 '16 at 16:51

Active Oct 30 '16 at 08:30

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Liouville's theorem of complex analysis states that a bounded entire function is constant. This means that for a non-constant entire function $f$ there exists a sequence $z_n \in \mathbb{C}$ such that $|z_n| \to \infty$ and $f(z_n) \to \infty$. I am trying to understand if a sort of converse holds in the following sense: consider a closed set $S \subset \mathbb{C}$ ($S$ may be unbounded) such that $\mathbb{C} \setminus S$ is unbounded. Can one construct a non-constant entire function $f$ such that $f$ is bounded on $S$ ?

Comment: Cross-posted on MO here.

edited Apr 13 '17 at 12:58

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asked Oct 28 '16 at 16:51

SMS

Try the Maximum modulus principle. But see http://math.stackexchange.com/questions/894304/maximum-modulus-principle-implies-liouvilles-theorem. – lhf Oct 28 '16 at 17:14
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What if $S= {0}?$ – zhw. Oct 28 '16 at 17:21
@lhf What if the maximum is attained at some boundary point of $S$? – SMS Oct 28 '16 at 17:53
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Yes, for every curve $\gamma$ such that $\gamma(t) \to \infty$, there exists a function bounded everywhere except in the neighborhood of the curve, search for the Alice Roth's theorems on entire functions – reuns Oct 30 '16 at 09:49

A question around Liouville's Theorem

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