The vii study presents some of the fundamental results annexed to the Hermite Hadamard inequality. It aims at improving the role of distinct classes of convexity in the theory of Inequalities. This dissertation is devoted to sift out several refinements, generalizations, improvements and extensions of the Hermite Hadamard type inequalities for functions whose absolute values of first and second derivatives belong to class of convex functions. Eventually, as applications, the obtained results are employed for special means of real numbers. Explicit bounds are also being derived to versatile composite quadrature rules in terms of variety of functions belonging to different classes of convex functions. The analysis sets aside the determination of the partition required that would ensure that the accuracy of the result would be within a prescribed error tolerance. This way, this thesis caters a study of some inequalities analogous to the most acclaimed and basic Hadamard inequality. The study piles up interesting developments in this research field under a unified framework. This work will be of keenness to mathematical analysts, pure and applied mathematicians, physicists, engineers, computer scientists and other areas of science.