Hermite-Hadamard type integral inequalities for s-convex functions

Abstract

One of the impressive application of the theory of convex functions is to the study of classical inequalities. Here, we show that how the theory provides an elementary, elegant, and uni ed treatment of some of the best known inequalities in mathematics. The fundamental purpose of this thesis is to establish some new Hermite-Hadamard type integral inequalities associated to s-convex function for (2!+1) times di erentiable functions. We assume de nite integrable function that can be di erentiated up to (2!+1) times on closed interval [0; 1]. The integrable function that we assumed, has an isolated singularities on 0 and 1. So, it is an improper integral. Our rst purpose was to remove these isolated singularities. Henceforth, to remove these isolated singularities we solved this improper integral by famous integration technique namely as integration by parts. After, solving and making some substitution we observed that it has no singularities on 0 and 1. The improper integral turns into proper integral. Here, we also used Binomial expansion to write integrable function in a compact form. The result that we obtain, named as a lemma. Then, we associates that lemma with Hermite- Hadamard type integral inequalities for s-convex function. We introduced several new results associated to s-convex function and extended s-convex functions. We used some famous integral inequalities i.e. classical Hermite-Hadamard integral inequality, power mean's integral inequality, Holder's integral inequality and Jensen integral inequality in order to obtain new results. These famous integral inequalities helps us a lot to solve our problem related to s-convex function.