  Recently, Lima and Randrianarivony pointed out the role of the property
$(\beta)$ of Rolewicz in nonlinear quotient problems, and answered a
ten-year-old question of Bates, Johnson, Lindenstrauss, Preiss and Schechtman.
In the present paper, we prove that the modulus of asymptotic uniform
smoothness of the range space of a uniform quotient map can be compared with
the modulus of $(\beta)$ of the domain space. We also provide conditions under
which this comparison can be improved.
