  Given a linear map $T$ on a Euclidean Jordan algebra of rank $n$, we consider
the set of all nonnegative vectors $q$ in $R^n$ with decreasing components that
satisfy the pointwise weak-majorization inequality
$\lambda(|T(x)|)\underset{w}{\prec}q*\lambda(|x|)$, where $\lambda$ is the
eigenvalue map and $*$ denotes the componentwise product in $R^n$. With respect
to the weak-majorization ordering, we show the existence of the least vector in
this set. When $T$ is a positive map, the least vector is shown to be the join
(in the weak-majorization order) of eigenvalue vectors of $T(e)$ and $T^*(e)$,
where $e$ is the unit element of the algebra. These results are analogous to
the results of Bapat, proved in the setting of the space of all $n\times n$
complex matrices with singular value map in place of the eigenvalue map. They
also extend two recent results of Tao, Jeong, and Gowda proved for quadratic
representations and Schur product induced transformations. As an application,
we provide an estimate on the norm of a general linear map relative to spectral
norms.
