-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.

Extensions of different type parameterized inequalities for generalized [Formula: see text]-preinvex mappings via -fractional integrals.

The authors discover a general -fractional integral identity with multi-parameters for twice differentiable functions. By using this integral equation, the authors derive some new bounds on Hermite-Hadamard's and Simpson's inequalities for generalized [Formula: see text]-preinvex functions through -fractional integrals. By taking the special parameter values for various suitable choices of function , some interesting results are also obtained.