  We revisit several results on exponential integrability in probability spaces
and derive some new ones. In particular, we give a quantitative form of recent
results by Cianchi-Musil and Pick in the framework of Moser-Trudinger-type
inequalities, and recover Ivanisvili-Russell's inequality for the Gaussian
measure. One key ingredient is the use of a dual argument, which is new in this
context, that we also implement in the discrete setting of the Poisson measure
on integers.
