  Let $r_1,\ldots,r_s:\mathbb{Z}_{n\geqslant 0}\to\mathbb{C}$ be linearly
recurrent sequences whose associated eigenvalues have arguments in
$\pi\mathbb{Q}$ and let $F(z):=\sum_{n\geqslant 0}f(n)z^n$, where
$f(n)\in\{r_1(n),\ldots,$ $r_s(n)\}$ for each $n\geqslant 0$. We prove that if
$F(z)$ is bounded in a sector of its disk of convergence, it is a rational
function. This extends a very recent result of Tang and Wang, who gave the
analogous result when the sequence $f(n)$ takes on values of finitely many
polynomials.
