In this paper we consider simple inequalities involving iterated integrals in the inequality
(1.1) for functions when the function u in the both side of the inequality (1.1) are replaced by
the function w(u) and ϕ (u) for some functions w,ϕ and provide some retarded integral inequalities
involving iterated integrals. Some applications are also given to illustrate the usefulness of
our results.