  Let $K$ be a field of characteristic zero and $x$ a free variable. A
$K$-$\mathcal E$-derivation of $K[x]$ is a $K$-linear map of the form
$\operatorname{I}-\phi$ for some $K$-algebra endomorphism $\phi$ of $K[x]$,
where $\operatorname{I}$ denotes the identity map of $K[x]$. In this paper we
study the image of an ideal of $K[x]$ under some $K$-derivations and
$K$-$\mathcal E$-derivations of $K[x]$. We show that the LFED conjecture
proposed in [Z4] holds for all $K$-$\mathcal E$-derivations and all locally
finite $K$-derivations of $K[x]$. We also show that the LNED conjecture
proposed in [Z4] holds for all locally nilpotent $K$-derivations of $K[x]$, and
also for all locally nilpotent $K$-$\mathcal E$-derivations of $K[x]$ and the
ideals $uK[x]$ such that either $u=0$, or $\operatorname{deg}\, u\le 1$, or $u$
has at least one repeated root in the algebraic closure of $K$. As a
bi-product, the homogeneous Mathieu subspaces (Mathieu-Zhao spaces) of the
univariate polynomial algebra over an arbitrary field have also been
classified.
