  The $\psi$-operator for $(\phi, \Gamma)$-modules plays an important role in
the study of Iwasawa theory via Fontaine's big rings. In this note, we prove
several sharp estimates for the $\psi$-operator in the cyclotomic case. These
estimates immediately imply a number of sharp $p$-adic combinatorial
congruences, one of which extends the classical congruences of Fleck (1913) and
Weisman (1977).
