Sharp Frobenius error transfer for monotone subhomogeneous matrix functions

Let $f:[0,\infty)\to[0,\infty)$ be continuous and nondecreasing, with $f(x)/x$
nonincreasing on $(0,\infty)$. This class includes every nonnegative
nondecreasing concave function. We prove that ordinary relative Frobenius
approximation error transfers from $A$ to $f(A)$ without loss whenever
$0\preceq B\preceq A$ and $B$ has the prescribed low rank. No
operator-monotonicity assumption or stronger trace-deficit hypothesis is
needed. Without the order assumption, the relative excess increases by at most
a factor of two, and this constant is optimal even for operator-monotone power
functions. Both conclusions follow from scalar inequalities combined with the
spectral decomposition of $A$; the ordered result additionally uses a harmonic
constraint on the eigenvectors of $B$.