  The stability properties of rotating relativistic stars against prompt
gravitational collapse to a black hole are rather well understood for uniformly
rotating models. This is not the case for differentially rotating neutron
stars, which are expected to be produced in catastrophic events such as the
merger of binary system of neutron stars or the collapse of a massive stellar
core. We consider sequences of differentially rotating equilibrium models using
the $j$-constant law and by combining them with their dynamical evolution, we
show that a sufficient stability criterion for differentially rotating neutron
stars exists similar to the one of their uniformly rotating counterparts.
Namely: along a sequence of constant angular momentum, a dynamical instability
sets in for central rest-mass densities slightly below the one of the
equilibrium solution at the turning point. In addition, following Breu &
Rezzolla (2016), we show that "quasi-universal" relations can be found when
calculating the turning-point mass. In turn, this allows us to compute the
maximum mass allowed by differential rotation, $M_{\rm
  max,dr}$, in terms of the maximum mass of the nonrotating configuration,
$M_{_{\rm TOV}}$, finding that $M_{\rm max, dr} \simeq \left(1.54 \pm
0.05\right) M_{_{\rm TOV}}$ for all the equations of state we have considered.
