In this paper, a new identity for $n$-times differntiable functions is established and by using the obtained identity, some new inequalities Hermite--Hadamard type are obtained for functions whose $n$th derivatives in absolute value are convex and concave functions. From our results, several inequalities of Hermite--Hadamard type can be derived in terms of functions whose first and second derivatives in absolute value are convex and concave functions as special cases. Our results may provide refinements of some results already exist in literature. Applications to trapezoidal formula and special means of established results are given.