Simpson’s type inequalities for exponentially convex functions with applications

Yenny Rangel-Oliveros,Eze R Nwaeze

doi:10.30538/psrp-oma2021.0096

Abstract

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.

Highlights

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
We show the importance of our results by applying them to some special means of positive real numbers and to the Simpson’s quadrature rule
The obtained results can be extended for different kinds of convex functions

Summary

Introduction

Simpson’s inequality for exponentially convex We give a new refinement of Simpson’s inequality for twice differentiable functions: Theorem 1. If |g |q is exponentially convex on [a1, a2] and q ≥ 1, the following inequality holds: q g (a2) q g (a1) q q. = g(a2) and |g |q is exponentially convex on [a1, a2] and q ≥ 1, the following inequality holds: Remark 3. Holds: Let g : I ⊂ R → R be a twice differentiable mapping on I◦ such that g ∈ L1[a1, a2], where.

Simpson’s inequality for exponentially quasi-convex

The logarithmic mean

Conclusion