Let $F : \mathbb C \to \mathbb C$ be an entire function of finite order. Since the zeros of $F$ are countable there exists a constant $c \in \mathbb R$ such that $F$ is zero-free on the line $e^{ic} \mathbb R$. I'm wondering of the following stronger statement holds true: there exists a $c \in \mathbb R$ and a $d>0$ such that
$$
|F(z)| \geq d
$$
for all $z \in e^{ic} \mathbb R$, that is, $F$ is uniformly bounded from below on some line $e^{ic} \mathbb R$. Above I wrote the assumption that $F$ is of finite order. I feel like such an assumption might be needed since it gives control over the density of the zeros of $F$.



Are entire functions uniformly bounded from below on a line through the origin?
The answer is negative, and here are two ways to construct counterexamples.


1. Let $G$ be a bounded simply connected region whose closure does not contain zero, but $G$ intersects every ray from the origin. For example, $G$ can be a neighborhood of a sufficiently long compact piece of a logarithmic spiral.
Consider the open set $D=\cup\_n t\_nG$, where $t\_n>0, t\_n\to\infty$ sufficiently fast, so that $t\_nG$ are disjoint. Then one can find an entire function $f$ such that $f(z)\to 0$ as $z\to\infty, z\in D$. Such a function
can be easily constructed using Runge's approximation theorem. Moreover, by taking the sequence $t\_n$ of sufficiently fast growth, one can control the growth of $f$, so that $f$ will be of finite order, and even of zero order. For details of the order control, one may consult, for example


MR1040926 Eremenko, A. E.; Sodin, M. L. On the behavior of an entire function on a sequence of concentric circles. Complex Variables Theory Appl. 12 (1989), no. 1-4, 267–276, or


MR0545054 Golʹdberg, A. A.; Eremenko, A. E. Asymptotic curves of entire functions of finite order. Mat. Sb. (N.S.) 109(151) (1979), no. 4, 555–581, 647 (there is an English translation).


2. One can use the construction of Balashov,


MR0324032
Balašov, S. K.
Functions with completely regular growth along curves of regular rotation.
Dokl. Akad. Nauk SSSR 209 (1973), 525–528 (There is an English translation).


MR0328073 Balašov, S. K. Entire functions of finite order with roots on curves of regular rotation. Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 603–629.
(There is an English translation).


(See also his papers MR0333171, MR0409807). Functions constructed in these papers are of any finite order $>1/2$ and have
zeros regularly distributed on spirals. They have controlled asymptotics on spirals, in particular one can arrange that $f(z)\to 0$ on some logarithmic spiral.


