## Integral inequalities of hermite-hadamard type and their applications

A thesis submitted to the Faculty of Science, University of the
Witwatersrand, Johannesburg, South Africa, in fulfilment of the
requirements for the degree of Doctor of Philosophy. Johannesburg, 17 October 2016. / The role of mathematical inequalities in the growth of different branches of mathematics
as well as in other areas of science is well recognized in the past several years. The uses of
contributions of Newton and Euler in mathematical analysis have resulted in a numerous
applications of modern mathematics in physical sciences, engineering and other areas
sciences and hence have employed a dominat effect on mathematical inequalities.
Mathematical inequalities play a dynamic role in numerical analysis for approximation of
errors in some quadrature rules. Speaking more specifically, the error approximation in
quadrature rules such as the mid-point rule, trapezoidal rule and Simpson rule etc. have
been investigated extensively and hence, a number of bounds for these quadrature rules in
terms of at most second derivative are proven by a number of researchers during the past
few years.
The theorey of mathematical inequalities heavily based on theory of convex functions.
Actually, the theory of convex functions is very old and its commencement is found to be
the end of the nineteenth century. The fundamental contributions of the theory of convex
functions can be found in the in the works of O. HÃ¶lder [50], O. Stolz [151] and J.
Hadamard [48]. At the beginning of the last century J. L. W. V. Jensen [72] first realized
the importance convex functions and commenced the symmetric study of the convex
functions. In years thereafter this research resulted in the appearance of the theory of
convex functions as an independent domain of mathematical analysis.
Although, there are a number of results based on convex function but the most celebrated
results about convex functions is the Hermite-Hadamard inequality, due to its rich
geometrical significance and many applications in the theory of means and in numerical
analysis. A huge number of research articles have been written during the last decade by a
number of mathematicians which give new proofs, generalizations, extensions and