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. Hölder [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 refitments of the Hermite-Hadamard inequality. Applications of the results for these classes of functions are given. The research upshots of this thesis make significant contributions in the theory of means and the theory of inequalities. / MT 2017

published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

Thesis (Ph. D.)--University of Hong Kong, 2009. / Includes bibliographical references (leaves 80-91). Also available in print.

In this thesis we derive various generalizations and refinements of some classical inequalities in different function spaces. We consider some of the most important inequalities namely the Hardy, Pólya-Knopp, Jensen, Minkowski and Beckenbach-Dresher inequalities. The main focus is put on the Hardy and their limit Pólya-Knopp inequalities. Indeed, we derive such inequalities even in a general Banach functionsetting. The thesis consists of three papers (A, B and C) and an introduction, which put these papers into a more general frame. This introduction has also independent interest since it shortly describe the dramatic more than 100 years of development of Hardy-type inequalities. It contains both well-known and very new ideas and results. In paper A we prove and discuss some new Hardy-type inequalities in Banach function space settings. In particular, such a result is proved and applied for a new general Hardy operator, which is introduced in this paper (this operator generalizes the usualHardy kernel operator). These results generalize and unify several classical Hardy-type inequalities. In paper B we prove some new refined Hardy-type inequalities again in Banach function space settings. The used (super quadraticity) technique is also illustrated by making refinements of some generalized forms of the Jensen, Minkowski and Beckenbach-Dresher inequalities. These results both generalize and unify several results of this type. In paper C for the case 0<p≤q<∞ we prove some new Pólya-Knopp inequalities in two and higher dimensions with good two-sided estimates of the sharp constants. By using this result and complementary ideas it is also proved a new multidimensional weighted Pólya-Knopp inequality with sharp constant. / In this thesis we derive various generalizations and refinements of some classical inequalities in different function spaces. We consider some of the most important inequalities namely the Hardy, Pólya-Knopp, Jensen, Minkowski and Beckenbach-Dresher inequalities. The main focus is put on the Hardy and their limit, Pólya-Knopp inequalities. Indeed, we derive such inequalities even in a general Banach function setting.  We prove and discuss some new Hardy-type inequalities in Banach function space settings. In particular, such a result is proved and applied for a new general Hardy operator. These results generalize and unify several classical Hardy-type inequalities. Next, we prove some new refined Hardy-type inequalities again in Banach function space settings. We used superquadraticity technique to prove refinements of some classical inequalities. Finally, for the case 0<p≤q<∞, we prove some new Pólya-Knopp inequalities in two and higher dimensions with good two-sided estimates of the sharp constants. By using this result and complementary ideas it is also proved a new multidimensional weighted Pólya-Knopp inequality with sharp constant.