  Exact lower bounds on the exponential moments of min(y,X) and XI{X<y} are
provided given the first two moments of a random variable X. These bounds are
useful in work on large deviations probabilities and nonuniform Berry-Esseen
bounds, when the Cram\'er tilt transform may be employed. Asymptotic properties
of these lower bounds are presented. Comparative advantages of the
Winsorization min(y,X) over the truncation XI{X<y} are demonstrated.
