  In this paper we study the question how to easily verify that the expectation
of the supremum of a one canonical Bernoulli process dominates the same
quantity for another process of this type. In the setting of Gaussian canonical
processes it is known that such a comparison holds for contractions in the
Euclidean distance. We do state and prove a similar result for Bernoulli
processes. In particular we get a partial answer to the Oleszkiewicz conjecture
about the comparability of weak and strong moments for type Bernoulli series in
a Banach space.
