GENERALIZATION OF HERMITE-HADAMARD TYPE INEQUALITIES AND THEIR APPLICATIONS

Abstract

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 for functions whose derivatives in absolute value raised to certain power are various classes of convexities. 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.