In this article, we define and consider some new concepts of the higher order exponentially general convex functions with respect to an arbitrary function. Some properties of the exponentially general convex functions are investigated under suitable conditions. It is shown that the optimality conditions of the higher order exponentially general convex functions are characterized by a class of variational inequalities, which is called the higher order exponentially variational inequality. Auxiliary principle technique is used to suggest some implicit method for solving exponentially general variational inequalities. Convergence analysis of the proposed method is investigated using the pseudomonotonicity of the operator. It is shown that the parallelogram laws for Banach spaces can be obtained as applications of higher order exponentially affine general convex functions. We have also discussed some properties of exponential general nonconvex variational inequalities applying the fixed point, Wiener–Hopf equations and dynamical system techniques. Some special cases also discussed. Results obtained in this paper can be viewed as refinement and improvement of previously known results.