-polynomial exponential type -convex function with some related inequalities and their applications.

In this paper, the idea and its algebraic properties of -polynomial exponential type -convex function have been investigated. Authors prove new trapezium type inequality for this new class of functions. We also obtain some refinements of the trapezium type inequality for functions whose first derivative in absolute value at certain power are -polynomial exponential type -convex. At the end, some new bounds for special means of different positive real numbers are provided as well. These new results yield us some generalizations of the prior results. Our idea and technique may stimulate further research in different areas of pure and applied sciences.