Simpson’s type inequalities for exponentially convex functions with applications

The Simpson's inequality cannot be applied to a function that is twice differentiable but not four times differentiable or have a bounded fourth derivative in the interval under consideration. Loads of articles are bound for twice differentiable convex functions but nothing, to the best of our knowledge, is known yet for twice differentiable exponentially convex and quasi-convex functions. In this paper, we aim to do justice to this query. For this, we prove several Simpson's type inequalities for exponentially convex and exponentially quasi-convex functions. Our findings refine, generalize and complement existing results in the literature. We regain previously known results by taking \(\alpha=0\). In addition, we also show the importance of our results by applying them to some special means of positive real numbers and to Simpson's quadrature rule. The obtained results can be extended for different kinds of convex functions.