$f \in H(\mathbb C) $ s.t. restricted to any strip of finite width ( including straight lines ) , $f(z) \to 0$ as $ z \to \infty$ ; is $f$ constant?

Question

Let $f:\mathbb C \to \mathbb C$ be an entire function . It is known that it is possible to have non-constant $f$ with the property that $f(z)\to 0$ as $z \to \infty$ restricted on any straight line . My question is : Does there exist a non-constant entire function $f:\mathbb C \to \mathbb C$ such that when restricted to any strip of finite width ( zero width hence just straight lines are allowed also ) , $f(z) \to 0$ as $ z \to \infty$ ?

Answer 1

Yes, such functions exist and can be constructed using Arakelyan's approximation theorem. Let $U = \{ x + iy : x \in (-1,\infty), y \in (x^2 - 1, x^2 + 1) \}$ (something like the $1$-neighborhood of the graph of $y = x^2$ for $x>-1$), and let $E = \mathbb{C} \setminus U$ be its complement. Note that every strip of finite width intersects $U$ in a bounded set, and that $E$ contains $1$, but not $0$.

Arakelyan's theorem gives that every continuous function on $E$ which is holomorphic in the interior can be uniformly approximated by entire functions. The function $g(z) = 1/z$ satisfies these assumptions and is bounded on $E$, so there exists an entire function $f$ with $|f(z) - 1/z| < 1/2$ for $z \in E$, which both implies that $f$ is bounded and that is non-constant on $E$. This already shows that there is a non-constant entire function $f$ which is bounded on every strip of finite width.

In order to improve this to get $\lim_{z\to\infty} f(z) = 0$ on every such strip, we have to modify the construction slightly. Note that $E$ is simply connected and does not contain $0$, so there exists an analytic branch of $-\log z$ in (some open neighborhood of) $E$. Applying Arakelyan's theorem to this function gives an entire function $h = u + iv$ with $|h(z) + \log z| < 1$ for $z \in E$. This shows that for $z \in E$ one has $u(z) < 1-\log |z|$ and thus $|e^{h(z)}| < e/|z|$, which implies that $\lim_{z\to\infty} h(z) = 0$ in $E$. So the entire function $f = e^h$ has the desired property.