  Lederer and van de Geer (2013) introduced a new Orlicz norm, the
Bernstein-Orlicz norm, which is connected to Bernstein type inequalities. Here
we introduce another Orlicz norm, the Bennett-Orlicz norm, which is connected
to Bennett type inequalities. The new Bennett-Orlicz norm yields inequalities
for expectations of maxima which are potentially somewhat tighter than those
resulting from the Bernstein-Orlicz norm when they are both applicable. We
discuss cross connections between these norms, exponential inequalities of the
Bernstein, Bennett, and Prokhorov types, and make comparisons with results of
Talagrand (1989, 1994), and Boucheron, Lugosi, and Massart (2013).
