Abstract
In this article, we introduce a new class of functions named the m-polynomial exponentially s-type convex and we study some of its algebraic properties. We investigate a new integral inequality of Hermite-Hadamard (H-H) type by using the new introduced definition. Also, we prove a new midpoint identity. By examining this identity we can deduce some integral inequalities of midpoint type for the new introduced definition. Several special cases are discussed in details which emphasize that the results accounted in this paper unify and extend various results in this field of study. Finally, we provide some applications on the Bessel functions and special means of distinct positive real numbers to demonstrate the applicability of the new results.