= Solution
Use the future-directed <null vector> $k=-\partial_r$, whose dual is $k_a=-(dv)_a$. The <stress-energy tensor> has the <null dust> form
$$
T_{ab}=\varepsilon k_ak_b,\qquad \varepsilon=\frac{\dot m(v)}{4\pi r^2}.
$$
For every future-directed <timelike vector> $t$, $k\cdot t<0$. The energy current appearing in the <dominant energy condition> is
$$
J^a=-T^a{}_bt^b=\varepsilon(-k\cdot t)k^a.
$$
If $\varepsilon\geq0$, this is a future-directed <null vector>, or zero; also $T_{ab}t^at^b=\varepsilon(k\cdot t)^2\geq0$. If $\varepsilon<0$, the current is past-directed and that energy density is negative. Therefore, given the already stipulated positivity and smoothness of $m$, the necessary and sufficient additional condition is
$$
\boxed{\dot m(v)\geq0\quad\text{for every }v.}
$$
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