| Resumo: | This thesis is concerned with the study of generalization, refinement, improvement
and extension of Hermite-Hadamard (H-H) type inequalities. These are achieved by
using various classes of convex functions and different fractional integrals. We established
new integral inequalities of H-H type via s-convex functions in the second
sense, as well as the new classes of convexities: h-Godunova-Levin and h-Godunova-
Levin preinvex functions. We also generalized the inequalities of the H-H type involving
Riemann-Liouville via generalized s-convex functions in the second sense
on fractal sets. We further generalized the H-H type inequalities involving Katugampola
fractional integrals via different types of convexities. We also improved several
inequalities of H-H type through various classes of convexities by using the conditions
| g' |q and | g" |q for q ≥ 1. Using the obtained new results, we presented
some applications to special means and applications to numerical integration. By
comparing the error bounds estimation of numerical integrations, report shows that
the present results obtained using generalization of mid-point and trapezoid type inequalities
are more efficient. Several quadrature rules were reported to be examined
through this approach. The findings of this study are new, more general and to some
extend better than many other research results.
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