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.
